Investigating the evolution of rock discontinuity asperity degradationand void space morphology under direct shear

Bryan Stanley Anthony Tatone

Abstract

Rock mass discontinuities represent planes of relative weakness and enhanced hydraulic conductivity and, thus, have a sub-stantial influence on the hydro-mechanical behaviour of the overall rock mass. While the shearing of rock mass discontinuitieshas been extensively studied in the past, there remains uncertainty surrounding the mechanisms by which surface asperitiesdeform and degrade during shear and how this degradation influences the aperture distribution. Although prior studies haveattempted to investigate asperity failure mechanisms, they have been hampered by the lack of appropriate visualization andmodelling tools. In particular, until recently it was not possible to observe asperity damage without physically separating thejoint specimen or explicitly modelling the development of damage during a direct shear test.

Over the last decade, micro X-ray Computed Tomography (µCT) has emerged as an ideal tool to nondestructively char-acterize fractures and damage in geomaterials. Over this same period, hybrid continuum/discontiuum modelling techniques,capable of explicitly modelling fracture and fragmentation have been developed and applied to rock mechanics problems.However, to date, there has been limited application of these technologies to the study of rock discontinuities subjected toshearing. The overall goal set forth in this thesis was to combine the use of these two technologies to develop an improvedunderstanding and confirm empirical assumptions regarding the evolution of asperity degradation and fracture geometry as aresult of shearing. The adopted experimental approach involved creating a series of replicated discontinuity specimens thatwere then subjected to varying shear displacements under different normal loading conditions. Subsequently, µCT imagery ofthe specimens was acquired and an image processing and analysis procedure was developed to quantitatively evaluate changesin asperity damage and fracture geometry as a function of shear displacement and applied normal load. Through the useof numerical modelling based on the hybrid Finite-Discrete element method (FDEM), the experimentally observed shearingprocess was then recreated numerically to glean further insight into the shearing process and the different failure mechanismsinvolved. Ultimately, this work has led to an improved understanding discontinuity morphology related to shearing, the failuremechanics of individual asperities, and the limitations of using a 2D FDEM approach to model discontinuity shearing.

1. Introduction

1.1. Problem statement

Developing an understanding of the shear behaviour of in-dividual discontinuities, in terms of shear strength and theability to convey fluids, has been the focus of a significantamount of research. Over the last four decades, numerousoriginal and modified shear strength criteria for rock disconti-nuities have been proposed to aid in rock engineering design[1, 2, 3, 4, 5, 6, 7, 8, amongst several others]. Similarly, nu-merous studies regarding the influence of shear displacement,discontinuity roughness, the degree of matching between thejoint walls, and the presence of gouge material or other fillingshave been undertaken [9, 10, 11, 12, 13, amongst several oth-ers]. This large quantity of research not only indicates the keyimportance of discontinuity shear strength and hydraulic be-havior in rock engineering and hydrogeology, but also demon-strates that the characterization of shear resistance and other

properties of rough discontinuities is a complex problem un-dergoing continued investigation.

Although the strength and hydraulic behaviour of discon-tinues have been extensively studied, a topic that has receivedlesser attention is the collective influence of shear displace-ment and normal stress on asperity damage and the over-all fracture morphology. In particular, there remains uncer-tainty regarding the kinematic mechanisms responsible forthe degradation of asperities and the influence of this damageon the discontinuity geometry (e.g., volume, aperture, surfacearea, and tortuosity). An improved understanding of thesemechanisms will be of value to several areas of rock engineer-ing and hydrogeology, including those concerned with pre-venting shear displacement (e.g., excavation, slope, and damstability analyses) and those concerned with changes in hy-draulic transmissivity resulting from shear displacement (e.g.,long-term radioactive waste repositories and reservoir stimu-lation by hydraulic fracturing).

PhD thesis summary December 28, 2015

1.2. State of the art and limitations of previous studies

For a rough rock discontinuity subjected to a constant aver-age normal stress, the typical shear resistance-shear displace-ment and the normal displacement-shear displacement (dila-tion) response can be divided into five phases, as shown inFig. 1a [after 14]:

• Phase I: Linearly-elastic deformation.

• Phase II: Shear stiffness reduction.

• Phase III: Peak shear strength.

• Phase IV: Post-peak softening.

• Phase V: Residual shear strength.

It is widely recognized that the transition from Phase 2 toPhase 4 involves a combination of asperity sliding and degra-dation with the amount of sliding or degradation related tothe magnitude of the applied normal stress (Fig. 1b). How-ever, examples of experimental or numerical studies that havespecifically focused on investigating asperity degradation, as-perity failure mechanisms, and the overall discontinuity mor-phology have been limited [e.g., 15, 16, 17, 18, 19, 20, 21,22, 23]. More importantly, these studies have been hinderedin one or more of the following ways:

• Limited discontinuity geometries. Many studies haveonly considered the behaviour of single asperities [e.g.,16, 20]) or idealized sawtooth geometries [e.g., 15, 18].Although it is recognized that one must gain an under-standing of the failure mechanisms of a single asperitiesand simple geometries before the behaviour of an assem-blage of natural asperities can be gained, considerationof a single asperities or idealized sawtooth patterns doesnot enable the progressive nature of asperity degradationto be studied.

• Weak replica material. All studies, except that of [18]that used machined rock, used a replica material suchas gypsum or cement mortar to replicate asperities en-abling identical test specimens to be rapidly produced.Although these materials are generally weaker than nat-ural rock, some of the materials used in prior studieshave been extremely weak (e.g., uniaxial compressivestrength < 10 MPa, tensile strength < 1 MPa) [e.g.,15, 16, 19, 20]. Thus, the assumption that replicated sur-faces are representative of rock discontinuities could berightly questioned.

• Limited and inconsistent reporting. In many previousstudies, several geometries were examined under varyingnormal stress, yet only selected results were reported as’typical’ observations [e.g., 15, 16, 18]. Thus, there waslittle information regarding the variation of the observed

Fig. 1: Schematic illustration of shear behaviour of rough rock discontinu-ities [14, 1]: (a) five phases of shear resistance and dilation response ob-served when shearing rough rock joints under constant normal loads and (b)schematic diagram of a typical peak shear strength envelope for a rough rockjoint.

failure geometries and mechanisms. Furthermore, the re-ported failure patterns were only representative of a sin-gle shear displacement. Hence, the evolution of damagewas not well documented and has led to disproportion-ately large number of terms being used to describe fail-ure mechanisms, including: shearing, tensile cracking,crushing, rotation, abrasion wearing, ploughing, glid-ing, grinding, breaking, brittle failure, separation, andcut-off [e.g., 15, 16, 17, 18, 19, 20]. It is appreciatedthat many of these terms have likely been used to clas-

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sify the same general failure mechanism, yet the largenumber of terms is indicative of the complexity and lackof consensus regarding the process.

• Validity of modelling. To date, numerical studies havebeen preliminary and demonstrative. That is, the re-sults were qualitatively assessed with minimal compar-ison to actual damage and stress-displacement behaviourrecorded in the laboratory [e.g., 21, 23, 22]. Neverthe-less, these numerical simulations of direct shear showpromise in providing insight into the progressive asper-ity failure mechanisms.

The limitations of previous studies have led to recurringrecommendations in literature for further study of discontinu-ities [e.g., 24, 25, amongst others ]. In particular, the studyof: individual asperity failure mechanisms, changes in thespatial aperture distribution (and contact area), influences ofgouge material, and the redistribution of local stresses throughall phases of the shearing process have been noted as requir-ing attention. Yet despite such recommendations, these topicshave remained largely unstudied due to the lack of appropri-ate tools and methodologies to adequately investigate the pro-cess. Until recently, it was difficult to experimentally measureand characterize the changes in surface morphology withoutphysically separating the two walls of the fracture [26, 27] orinjecting a hardening liquid to form a cast of the aperture orperform serial sectioning [e.g., 28, 29]. Similarly, not untilthe last decade have numerical modelling approaches capableof explicitly simulating the fracture and fragmentation char-acteristic of sheared discontinuities become widely available[e.g., 30, 31, 32]. However, now with both improved ex-perimental and numerical tools widely available, new studiesaimed at characterizing discontinuity shearing behaviour canproceed.

1.3. Overview of adopted approachFrom an experimental perspective, micro X-ray computed

tomography (µCT) has emerged over the last decade as an ef-fective approach to characterize the internal structure of rockspecimens. CT scanning captures the internal variation ofX-ray attenuation which is directly related to material thick-ness, density, and elemental composition. Considering thesharp density contrast between intact rock and void space (i.e.,pores and fractures), CT is ideally suited to imaging these fea-tures without physically opening the joint [33]. Although avariety of data processing methodologies have become well-established to qualitatively and quantitatively investigate jointapertures that are less than 100 µm [e.g., 34, 35] and charac-terize damage in rock specimens [e.g., 36, 37]), this technol-ogy had yet to be used to characterize sheared rock disconti-nuities before starting this thesis.

From a numerical perspective, modelling based on hybridfinite-discrete element method (FDEM) capable of explicitlyaccounting for the interaction of discrete bodies with rough

surfaces, fracture initiation, and fracture propagation are nowavailable. Tools such as Y-Geo FDEM [32, 38] permit thecomplete stress-displacement response to be modelled as anemergent behaviour, eliminating the need for an empiricalconstitutive failure model [23, 30]. That is, a complex non-linear macroscopic response can be reproduced by a collec-tion of simple micro-scale behaviours. Despite the potentialto explicitly simulate the dilation and asperity degradation ofdiscontinues directly from modelling output, the applicationof these techniques to simulate discontinuity shearing wasminimal prior to starting this thesis.

Given the availability of the above technologies, the re-search methodology adopted for the current research aimedto overcome many of the limitations of previous studies byadopting a two-facetted approach, including experimental andnumerical components. Prior to this work, cross-over be-tween numerical and experimental approaches to study asper-ity damage and void space evolution has been limited. That is,experimental studies have rarely considered numerical mod-elling results and numerical studies have rarely consideredlaboratory results to validate and corroborate the interpreta-tion of their respective results. Bridging this gap involvedcombining direct shear testing and CT image analysis of dis-continuity replicas with FDEM numerical modelling to sim-ulate direct shear tests (Fig. 2). By using these technologiestogether, asperity damage and fracture aperture could be di-rectly observed, measured, and explicitly modelled for thefirst time. In particular, the CT imagery of the sheared spec-imens was used to validate the numerically simulated sheardisplacement response and associated damage patterns. Andat the same time, the numerically simulated stress conditionsand fracture modes were used to help interpret the failuremechanisms responsible for damage observed in the shearedspecimens. Ultimately, by considering the experimental andnumerical results together new insights regarding rock discon-tinuity asperity damage evolution and void space morphologywere obtained and the limitations of using FDEM to modeldiscontinuity shearing were clearly identified.

2. Theoretical and practical advancements in rock me-chanics

When pursuing the ultimate goal of gaining an improvedunderstanding of discontinuity asperity degradation and voidspace morphology according to the work flow defined inFig. 2, several experimental and numerical advancementswere realized. These advancements are highlighted in the fol-lowing subsections.

2.1. Advancement in replica material selection

The investigation of asperity degradation and failure mech-anisms using specimens of natural rock joints is compli-cated by the large number of potential variables influenc-ing the process and their natural variability, including: joint

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Fig. 2: Overview of two-facetted work flow.

wall strength, grain scale heterogeneity (mineralogy, size, andshape), anisotropic 3D joint roughness, normal boundary con-ditions, and specimen size. The replication of rock joints us-ing a plaster-based or cement-based ‘rock-like’ material hasbeen a common approach to reduce the number of variablesinfluencing the rock joint shearing process. Several studieshave used such materials to aid in the study of the differ-ent aspects of discontinuity shear shear and asperity failure[15, 16, 17, 18, 19, 20, 39, 40, 41].

To characterize asperity degradation with µCT imagery andFDEM modelling in the current research, replicated joint sur-faces were also utilized. However, before creating these jointreplicas, an appropriate replica material needed to be selected.The employed mortar was a commercially available productcomprised primarily of a blend of sulfate cement and calciumcarbonate. The material is manufactured by King ConcreteProducts and retails in Canada under the name Flowstone.The mechanical properties of the cured material, includingthe UCS, BDS, triaxial strength envelope, and basic frictionangle were characterized through standard laboratory testing(Table 1).

Although the mechanical behavior of a man-made materialwill always differ from that of a natural rock, some materialsare mechanically more rock-like than others. One of the short-comings of many prior studies was the use of relatively weakreplica materials or materials with UCS to elastic modulus,E, and UCS to Brazilian disc strength (BDS) ratios that felloutside the normal range for rock. In contrast, the mechanicalproperties of the micro-fine mortar material employed for thecurrent work were more like those characteristic of a lime-stone.

Considering the E:UCS sedimentary rock classificationchart for the classification of sedimentary rocks [42], manydata points representing synthetic rock materials from priorstudies over the last three decades plot well outside the rangestypical of natural rocks (Fig. 3a). In contrast, the mortar

Table 1: Summary of material properties of mortar material selected for rockfracture replication.

Material property Mean value±std. dev.

Water:binder ratio 0.3Nominal grain size (µm) 3-5Indirect tensile strength (MPa) 2.6±0.53Unconfined compressive strength (MPa) 50.3±4.17Elastic modulus (GPa) 15.2±0.48Poisson’s ratio 0.25±0.004Dry density (g/cm3) 1.7±0.021Internal friction angle (◦) 22.9Cohesion (MPa) 16.4Basic friction angle (◦) 25.4±3.4

adopted in the current study had a E:UCS falling well withinthe average range for natural sedimentary rocks.

Considering the UCS:BD ratio of replica materials, a toollike the classification chart of [42] was not available. There-fore as part of this thesis, an extensive literature search wasconducted to develop a database of BD and UCS values forseveral rock types. Over 400 pairs of UCS and BD strengthvalues were tabulated, including values for multiple sedimen-tary, igneous, metamorphic rock types. Using a chart lay-out similar to that adopted by [42], the pairs of UCS and BDvalues were plotted in a log-log plot. In doing so, a high,low, and average range of the UCS:BD ratio for rocks couldbe established and different rock types produced clusters thatcould be statically identified. When points representing dif-ferent synthetic rock materials, including the mortar adoptedin this thesis, were plotted on the sedimentary chart (Fig. 3b),it was again evident the selected material was more similar tonatural sedimentary rock types than many previously adoptedmaterials.

To further assess the selected mortar material, the Poisson’sratio and density were compared to values for a variety of rocktypes (Fig. 4). While the Poisson’s ratio of 0.24 lies withinthe range of measured values for nearly all rock types, the drydensity of 1.7 g/cm3 is lower than that for most rocks types,but within the range defined by some sedimentary rocks.

2.2. Experimental advances

To examine the evolution of asperity degradation exper-imentally via µCT imaging, several experimental advanceswere realized, including: the design of discontinuity speci-mens with a geometry ideal for imaging, the development ofa direct shear testing procedure, and developing the method-ology and software tools required to process and analyze theµCT imagery to obtain quantitative measurements.

2.2.1. Specimen replicationConventionally, direct shear testing of discontinuities in-

volves imposing shear displacement on prismatic samples viaa shear box. CT scanning of prismatic samples, however, can

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Fig. 3: Comparison of the modulus and strength ratios of synthetic rock material to natural sedimentary rocks: (a) classification based on modulus ratio (after[42]); (b) classification based on strength ratio.

Fig. 4: Comparison of the Poisson’s ratio and density of Flowstone to natural rocks: (a) range of values for Poisson’s ratio for several rock types (after [43]);(b) range of dry densities for several rock types (data from [44, 45]).

be problematic in that artifacts can arise in the reconstructedvolume. As a conventional prismatic direct shear specimenis rotated and irradiated with X-rays from different angles,the apparent thickness of the specimen varies and, thus theray paths through the specimen are of unequal length. Thisvariation in apparent thickness leads to variations in X-rayattenuation that can manifest themselves as streaks in the re-constructed 3D volume of the specimen [46]. With cylindri-cal specimens, such artifacts are avoided and variations in the

reconstructed 3D X-ray attenuation can be attributed solelyto differences in the internal density and composition of thespecimen. Therefore, cylindrical direct shear specimens wereadopted for the present study.

A suite of direct shear specimens with idealized and natu-ral discontinuity surfaces were created using the mortar mate-rial described previously in Section 2.1 (Fig. 5). These spec-imens, which measured 54 mm in diameter x 83 mm long,included a longitudinal discontinuity surface and a flat, 5 mm

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deep, recess around the perimeter of the discontinuity surfaceto facilitate bonding the specimen halves together followingdirect shear testing. The various discontinuity surface geome-tries were cast using urethane rubber molds while the overallcylindrical shape was achieved by casting within Delrin splitmolds with the desired diameter. These molds were filled ona vibrating table, which proved to be essential in reducing thepresence of entrapped air bubbles in the replicas.

Six replicas of each geometry were created such that speci-mens of each geometry could be sheared to three different dis-placements under two different normal loads. Hence, in total,36 direct shear tests were completed as part of this study.

2.2.2. Direct shear testing and µCT image acquisitionBy replacing the standard square shear box of a direct shear

test apparatus with a newly-designed and fabricated semi-circular specimen holder (Fig. 6a), the specimens were testedusing a simple electro-mechanical direct shear apparatus(Fig. 6b). Tests were completed under constant normal load-ing conditions with a displacement rate of 0.005 mm/s. Foreach specimen geometry, separate specimens were sheared to6 mm, 3 mm, and to a displacement corresponding to the peakshear resistance. Once the desired incremental shear displace-ment was reached for a given specimen, and before the normaland residual shear load was removed, the 5 mm recess aroundthe perimeter of the discontinuity surface was filled with aquick setting epoxy adhesive to effectively bond the halves ofthe specimen together. By doing so, the relative position ofthe two specimen halves could be maintained while relocat-ing the specimen to a µCT cabinet for scanning. Althoughit is recognized that the residual shear and normal stressescould not be perfectly maintained on the specimens with this

2 mm

2 mm

4 mm

4 mm

8 mm

4 mm

8 mm

Fig. 5: Replicated saw-tooth and natural discontinuity surfaces.

approach, changes in specimen geometry due to elastic relax-ation were deemed insignificant compared to those resultingfrom shear-induced inelastic deformation.

The micro-CT scanner used for the present study was aPhoenix X-ray V|tome|x system manufactured by GeneralElectric Sensing and Inspection Technologies and is locatedwithin the Department of Civil Engineering at the Univer-sity of Toronto (Fig. 7a). This system includes an acquisitioncomputer; a reconstruction computer; and a shielded cabinet,which houses a 240 kV micro-focus X-ray tube, an X-ray de-tector array coupled to a 1024 × 1024 pixel charge-coupleddevice (CCD) sensor, and a 5-axis Computer Numerical Con-trol (CNC) rotation stage to hold the direct shear specimen tobe scanned (Fig. 7b).

The settings used to scan the direct shear specimens aresummarized in Table 2. Each specimen was irradiated with X-

(a)(a) (b)(b) 10 mm displacement 10 mm displacement transducerstransducers

13 13 kNkNload cellload cell Direct shearDirect shear

specimenspecimen

(c)(c)

25 mm25 mmXX bb

Rotation Rotation 

DetectorDetector

displacementdisplacementtransducertransducer

XX‐‐ray tuberay tubestagestage

Fig. 6: Direct shear testing apparatus: (a) newly-designed specimen holderfor cylindrical specimens and (b) view of entire apparatus.

(a)(a)

(b)(b)

Direct shearDirect shearspecimenspecimen

RotationRotation

DetectorDetector

specimenspecimen

XX‐‐ray tuberay tubeRotation Rotation stagestage

Fig. 7: (a) Phoenix V|tome|x micro-CT scanner at the University of Torontoand (b) main system components inside micro-CT cabinet.

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rays over its circumference by rotating it 360◦ in 1080 equallyspaced increments. At each position, 10 individual projec-tions were acquired and averaged to obtain the 2D 16-bit greyscale projection to be used for reconstruction. Maximizing themagnification of the specimens within the field of view corre-sponded to a voxel resolution of 100 µm (i.e., voxels measured100 µm × 100 µm × 100 µm). Reconstruction of the 3D imagevia filtered back projection was performed using the PhoenixX-ray Datos|x-reconstruction software (v. 1.5.0.22).

An example of the incremental direct shear testing resultsand the corresponding µCT-derived specimen damage evolu-tion for a serpentinite discontinuity geometry subjected to anormal load of 3 kN are shown in Fig. 8. Such images aresome of the first showing discontinuity morphology within aspecimen during direct shearing. Also note that the overlapof the stress-displacement and dilation-displacement curvesand position of damage localization is indicative of the con-sistency of the strength and geometry of the replicated testspecimens.

2.2.3. µCT image processing and analysisTo move from qualitative assessments of the discontinuity

morphology to more quantitative characterization, additionalµCT image processing and analysis was needed. In this the-sis, many existing tools and plug-ins within the open-sourcesoftware FIJI along with several newly created plug-ins andmacros were utilized to do so [47]. At the time of completion,this marked the first known attempt at quantitatively charac-terizing such features via µCT imaging.

In general, the image processing and analysis involved thefollowing steps:

1. Image preprocessing - The reconstructed 16-bit imagesof the shear specimens were aligned such that the dis-continuity plane was perpendicular to the slices of thevolume. Each volume was then cropped down to a re-gion encompassing the discontinuities with the same di-mensions.

2. Segmentation - A segmentation algorithm was imple-mented as a new plugin for FIJI called ShearSeg3D. Theapproach involved coupling dual-level grey value thresh-olds with a spatial second derivative threshold. The dual-

Table 2: Micro-CT settings used to scan direct shear specimens.

Parameter Value

Number of images 1080Voltage (mV) 140Current (µA) 160Detector timing (ms) 600Image averaging 10Voxel resolution (µm) 100 × 100 × 100Pre filtering 0.5 mm Cu

level global thresholds were used to obtain an approxi-mate segmentation from the image histogram and obtaina ternary image (e.g., Fig. 9a, b and e) (referred to as theprimary segmentation) and the second derivative thresh-old, a method typically used for edge detection [48, 49],was employed to refine the primary segmentation (re-ferred to as the secondary segmentation) (Fig. 9c).

3. Morphological operations - To improve the binary im-ages for the purpose of performing image-based mea-surements, noise treatment, hole-filling, and a watershedalgorithm were applied. These operations provided ahigh quality binary image stack suitable for quantitativemeasurements (Fig. 9d).

4. Image-based measurements - Lastly, a series of newmacro and plugins for FIJI were developed to make bi-nary image-based measurements, including: mean frac-ture aperture, fracture surface area, median effectiveaperture, spatial distribution of effective aperture, and thepreferred orientation of the discontinuity void space.

The spatial distribution of effective fracture aperture (e.g.,Fig. 10a) was defined by calculating the sum of void voxelsin a direction normal to the shear plane. The statistical distri-bution (e.g., Fig. 10b) was obtained by simply considering alleffective aperture measurements across the discontinuity. Asshown in the figure, only select areas of the discontinuity ex-perience an increase in aperture with shear displacement. Dueto the anisotropic roughness of the surface, channels perpen-dicular to the shear direction can be clearly observed (e.g.,Fig. 10a). The increase in aperture is reflected by the me-dian effective, which increased by nearly 300 % from the peakshear displacement to a final displacement of 6 mm (Fig. 5a).Over the same displacement range, the mean fracture aper-ture, estimated by dividing the total volume of void spaceby the cross-sectional area of the discontinuity specimens, in-creased by a similar magnitude (Fig. 5a). The total fracturesurface area, defined by the interfacial area between intact ma-terials and void space, showed little change as a function ofdisplacement for this particular specimen due to the minorareal extent of asperity damage. (Fig. 11a)

The anisotropy in the preferential orientation of fracturevoid space was characterized in terms of a Mean InterceptLength (MIL). By overlaying a series of parallel lines overthe discontinuity-parallel binary images, the mean length ofthe segments that intercept the void space was calculated. Alarge value for the MIL, indicates that the void space is morecontinuous, while a small value indicates that the void space ismore discontinuous. Through consideration of parallel linesin many directions, the continuity of the void space as a func-tion of direction was characterized. Such data can be pre-sented in a polar plot (Fig. 11b). The MIL values for theserpentinite fracture indicate that the void space is preferen-tially oriented perpendicular to the shearing direction and be-comes more continuous with increases in shear displacement(in agreement with Fig. 10).

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Fig. 8: Qualitative evaluation of of asperity morphology for replicated discontinuity specimens subjected to increasing direct shear displacements.

Fig. 10: (a) Evolution of the spatial effective aperture distribution for a serpentinite discontinuity surface as a function of shear displacement; (b) correspondingstatistical distributions of effective aperture.

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Fig. 9: (a) Initial grey-scale µCT image; (b) results of primary segmentation;(c) results of secondary segmentation; (d) binary image following morpho-logical operations; and (e) grey-scale histogram of initial image.

2.3. Numerical advances

A common limitation of conventional continuum (e.g.,finite element methods (FEM), finite difference methods(FDM), and boundary element methods (BEM)) and discon-tinuum (e.g., discrete element method (DEM) and discontin-uous deformation analysis (DDA)) numerical modelling ap-proaches is the inability to capture the formation of new dis-continuities generated by brittle fracturing processes. Al-though several continuum-based approaches that incorporateemergent discontinuities have been developed (e.g., [50, 51,52]), approaches based on “bonded” discrete element meth-ods have arguably become the most popular. In particular,bonded particle models (BPMs) within PFC2D/3D [53, 54]as well as bonded assemblages of triangular and Voronoi el-ements within UDEC [55] have been widely used to sim-ulate brittle fracturing in rock and other geomaterials (e.g.,[30, 56, 57, 58, 59, 60, 61, 62], among many others). In addi-tion to bonded-DEM approaches, the hybrid combined finite-discrete element method (FDEM) pioneered by [32, 63] and

implemented in the ELFEN [64] and Y-Geo [38] codes hasemerged as a promising technique to explicitly simulate frac-ture and fragmentation processes in brittle geomaterials (e.g.,[31, 65, 66]).

Fig. 11: (a) Changes in median effective aperture, mean aperture, and fracturesurface area as a function of shear displacement for a serpentinite disconti-nuity surface; (b) Mean intercept length measured parallel to the serpentinitediscontinuity surface at varying shear displacements.

With FDEM the dynamic simulation of multiple interact-ing bodies can be realized. A simulation can begin with asingle intact domain or a collection of discrete intact bodies.As the simulation progresses, these bodies can deform elas-tically, translate, rotate, interact, and fracture upon satisfyingsome fracture criterion, thus producing new discrete bodies.The newly generated bodies can then undergo further move-ment, interaction, deformation, and fracture. The approachemploys a combination of FEM techniques to assess the de-formation and evaluate the failure criterion for fracturing andDEM concepts for detecting new contacts and dealing withthe translation, rotation, and interaction of discrete bodies.

A key benefit of both the bonded DEM and FDEM ap-proaches is that failure evolution laws do not need to be pre-scribed. Instead, the macroscopic behaviour of the materialdevelops as a result of the evolution of micro-scale damage[30]. Nevertheless, this ability to capture more complex pro-

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cesses is directly linked with an increase in number of inputparameters which must be correctly specified. Unfortunately,many of these parameters, even those based directly on frac-ture mechanics principles, cannot be measured in the labora-tory. Instead, these parameters must be established throughsome calibration procedure. Generally the calibration of suchmodels involves adjusting the micro-mechanical parametersvia trial and error until the modelled macroscopic responsecorresponds to the observed response [30, 56, 57, 66]. Of-ten this number of parameters make the calibration processan over determined problem. As a result, the validity ofsuch modelling techniques are often rightly questioned. Toimprove the confidence in the results of such models, moreprescriptive calibration procedures must be devised to obtainsuitable input parameters [59, 60, 67, 68].

2.3.1. Calibration procedure developmentAs part of this thesis, a prescriptive calibration procedure

for laboratory-scale models in the 2D Y-Geo FDEM code [38]was developed [47]. Prior to this development, the selec-tion of appropriate input parameters for FDEM simulationswas accomplished through ad hoc approaches. The system-atic methodology represents a significant step towards reli-ably simulating lab-scale behaviour of rock and other brittlegeomaterials. To date, this work also represents the most ex-tensive illustration of the sensitivity of emergent strength andfailure modes/patterns to differing input parameters, meshes,and loading velocities.

The developed calibration procedure involves running se-ries of UCS, BD, and biaxial simulations to first identify andminimize mesh dependency and loading rate dependency andsubsequently derive a set of values for the input parameterscontrolling the intact material stiffness (penalty terms) andunconfined/confined material strength (Fig. 12). In followingthis procedure, a combination of input parameters that resultin emergent UCS and BD strengths and a simulated triaxialstrength envelope that closely match those measured in thelaboratory are obtained (Fig. 13). Moreover, these final cal-ibrated parameters result in fracture modes and patterns thatwere in agreement with those observed in laboratory speci-mens.

To simulate the direct testing of discontinuities, additionalcalibration efforts were required to determine values for theparameters controlling the behaviour of the discontinuityinterface, namely the friction angle and stiffness (penalty)terms. Considering a direct shear test simulation of a spec-imen with flat interface, a systematic approach to adjust wasdeveloped such that the emergent frictional resistance versusdisplacement matched that measured for a flat mortar replicain laboratory. In using the final calibrated input parameters,confidence in the results of subsequent forward modelling ofdirect shear specimens with other discontinuity geometrieswas greatly improved.

Fig. 12: Flowchart outlining calibration procedure.

3. Verification of the proposed solution

One of the central themes of this thesis relates to the bene-fits of using both novel experimental and numerical methodstogether to obtain new insights on a classic problem. Essen-tially, the findings from the experimental and numerical stud-ies were considered together in an attempt to cross-validatethe results of one another. Namely, experimental direct shearresults and µCT imaging were used to verify the emergentstrength and failure pattern obtained in FDEM simulationsof direct shear tests. And at the same time, the results ofthe FDEM simulations provided information about the failuremechanisms responsible for the damage patterns observed inµCT images.

The following subsections first describe the setup of the nu-merical direct shear tests and how the results compare with ex-perimental observations. Secondly, with the modelling limita-tions well-defined, the evolution of asperity damage and cor-responding stress conditions with increasing shear displace-ment are outlined. Lastly, based on this new understanding,the new conceptual model describing the progressive break-down of asperities is presented.

3.1. Direct shear tests: simulations versus reality

With the intact material properties assigned via the calibra-tion process, the laboratory direct shear simulations could be

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Fig. 13: Final calibration results for the mortar material used to replicatediscontinuity specimens.

simulated numerically. In doing so, the failure pattern, failuremode, and the associated stress conditions could be furtherexamined. To construct the 2D FDEM models, representative2D profiles were selected from the 3D discontinuity surfaces(e.g., Fig. 14a). These profiles were obtained from locations

that experienced the greatest concentrations of asperity dam-age in laboratory tests and, thus, dictated the overall directshear response. An example of a complete model contain-ing a 2D profile of the serpentinite discontinuity surface isillustrated in Fig. 14b. In this model, the 2D discontinuityprofile is represented by a series 0.75 mm long straight linesegments. By doing so, 0.75 mm triangular elements could beused to obtain a fine discretization of the specimen in the re-gion where damage was expected. The mesh was then gradedto larger element sizes away from the discontinuity interfaceto reduce the total number of elements required and, thus,computational demands.

In the direct shear models, an effort was made to reproducethe loading conditions imposed by the direct testing machinein the laboratory as closely as possible. Varying normal loadswere simulated by applying an element surface pressure alongthe central portion of the upper shear box (e.g., Fig. 14b). Thissurface pressure was applied in a gradual manner to quickly

(a) (a)

Profile 2

(b)

Profile 2

(b)

~7200 elements ~7200 elements

Fig. 14: (a) Example of location of 2D profiles on 3D serpentinite surface;(b) Example of 2D direct shear simulation geometry showing mesh topology,boundary conditions, and data output points for profile 2.

11

obtain a steady-state stress condition. Afterwards, a constantrate of shear displacement was applied to the lower half of thedirect shear box, while the upper half of the shear box washeld stationary in the horizontal plane. The output historypoints labelled 1 through 8 were used to monitor the shear-ing process, including the load-displacement, and dilation-displacement response.

Considering the serpentinite discontinuity (Fig. 14), anexample of the simulated shear resistance, in terms of theinstantaneous friction angle, and dilation as a function ofshear displacement is plotted in Fig. 15 alongside the corre-sponding laboratory results for comparison. The correspond-ing simulated, Mode-I-dominated asperity damage and voidspace morphology as a function of shear displacement is alsoshown.

With regard to the similitude between numerical and lab-oratory results, the peak shear resistance, dilation and crackpatterns were generally in good agreement for up to 2 mm ofshear displacement. However, the shear stiffness, post-peakshear resistance, and crack patterns at large displacementsvaried from that observed in laboratory specimens.

The most striking feature of the simulated shear resistance-displacement plot is the erratic nature of post-peak responsebeyond 2 mm of shear displacement. This response is at-tributed to unstable crack propagation events which rapidlyrelease built-up asperity stresses. In the laboratory, suchstresses are rapidly and nearly seamlessly redistributed toother contact areas over the width of the specimen. How-ever, in a 2D FDEM model, stresses can only be redistributedwithin the modelled cross-section. Thus, the progressive for-mation of new fractures can result in sudden losses of shear re-sistance until new asperity contacts are established with con-tinuing shear displacement. With the redistribution of stressesoccurring across the modelled cross-section only, it also fol-lows that asperity damage is restricted to the modeled cross-section. As a result, the degree of simulated asperity dam-age begins to exceed that observed in laboratory specimens atshear displacements beyond 2-3 mm.

3.2. Asperity damage evolution: Utilizing numerical and ex-perimental results together

Given the laboratory direct shear test results and µCT im-agery, it was possible to identify the limitations of adoptinga 2D FDEM approach to model discontinuity shearing. Mostimportantly, the µCT imagery helped avoid invalid interpre-tations of asperity damage mechanisms based on numericalresults alone. Given that the results of FDEM direct shear testsimulations were in good agreement with laboratory observa-tions over the first 2-3 mm of shear displacement, this portionof FDEM simulations could be used to help understand theevolution of asperity stress conditions and damage.

Consider the stresses within a group of elements in the cen-tre of one of the 2 mm ‘saw-tooth’ asperities tested under aconstant normal load of 2 kN are assessed (Fig. 16a-inset). By

computing the average major and minor principal stresses, σ1and σ3, for these elements over the initial 1 mm of simulatedshear displacement, the stress path up to failure for this asper-ity can be examined (Fig. 16a). The simulated crack patterns,principal stress distributions, and principal stress trajectoriesat points along this stress path are illustrated in Fig. 16c. Thegenerality of this simulated stress and asperity damage evo-lution is supported by the similar asperity crack patterns ob-served in the µCT imagery of discontinuities surfaces withvarying geometry under varying normal stresses (e.g., Fig. 8and Fig. 16b).

Upon initiating shear displacement, both principal stressesare compressive (Fig. 16a). However, with minimal displace-ment, σ3 begins to decrease and transition from a compressiveto tensile stress. As σ3 approaches the tensile strength of thematerial, the first tensile fractures initiate (Fig. 16b, i). Fol-lowing initiation, the stress path becomes cyclic with σ3 al-ternating between tensile and compressive stress states as theasperity progressively breaks down (Fig. 16b, iii-v). Uponcomplete failure of the asperity (e.g., Fig. 16b, vi), the princi-pal stresses within the selected elements go to zero.

Inspection of the principal stress trajectories helps explainthe orientation of fracturing (Fig. 16b, i-vi). Tensile frac-tures initially form parallel to the direction of σ1, which isperpendicular to the contacting interface (Fig. 16b, ii). Asthese initial cracks propagate, the stress trajectories ahead ofthe crack tip begin to rotate (Fig. 16b, iii-iv). This stressrotation ahead of the crack tip causes further crack propa-gation to follow a new trajectory that is approximately 120degrees counterclockwise from the initial fracture trajectory(Fig. 16b, v). This new crack trajectory directs the propagat-ing tensile fractures back towards the discontinuity interface.If the tensile fractures reach the discontinuity interface, theasperity becomes completely detached and subject to rotationand further breakdown with further shear displacement. Al-ternatively, shear displacement of crack elements lead to thedevelopment of shear fractures within the asperity. These in-dividual secondary shear fractures tend to link the existingtensile fractures together forming a pseudo-continuous shearplanes across the base of asperities.

3.3. A new conceptual model

Based on the experimental observations together with theresults of the 2D FDEM simulations of direct shear tests, ageneralized conceptual model for the degradation of an asper-ity subjected to a shear loading was developed (Fig. 17). Themodel is comprised of Stages 0 to 5, each of which bridge dif-ferent stages of a constant normal load direct shear test (i.e.,Fig. 1). The model is considered valid over an intermediaterange of constant normal loads where shear displacement isaccommodated by a combination of sliding and brittle frac-turing of asperities and dilation is permitted. This valid stressrange will vary according to the intact strength of the disconti-nuity walls. Descriptions of the stages of asperity degradation

12

NUMERICAL: DIRECT SHEAR SIMULATIONS

9Fig. 15: (a) Example of simulated direst shear test results for the serpentinite discontinuity geometry displayed in (Figure) and (b) Corresponding simulatedasperity damage evolution.

are as follows:

• Stage 0 This stage represents the initial configuration ofthe discontinuity prior to any shear displacement.

• Stage 1 As sliding initiates, frictional forces mobilizeparallel to the contacting area. These frictional forcesoppose the relative displacement and generate tensilestresses in the contacting asperities. When these tensilestresses exceed the intact strength of the asperity, a ten-sile crack perpendicular to the contact area and parallelto the maximum principal stress orientation initiates.

• Stage 2 With a minor increase in shear displacement, thefrictional forces opposing the shear displacement con-tinue to contribute to the overall shear resistance of thediscontinuity and, therefore, maintain the tensile stressstate in the asperities. Existing tensile cracks propagatein the same orientation and new parallel tensile fracturesinitiate as the tensile strength is overcome at additionallocations.

• Stage 3 Extension of the propagating tensile fracturesleads to the division of the asperity into multiple slen-der lengths of intact material. With continued shear dis-placement, the longer pieces of material have a tendencyto rotate. This rotation is accompanied by a rotation of

the principal stress directions ahead of the crack tips thatcause further fracture propagation to occur along a newtrajectory. Simultaneously, the shorter lengths of intactmaterial are compressed along their long axis, resultingin the formation of shear cracks that transect the asperi-ties.

• Stage 4 Extension of tensile fractures along the new tra-jectories continues until extending back to the originaldiscontinuity interface. At the same time, shear crackscontinue to link existing cracks together. The increasesin damage further reduce the ability of the asperity to re-sist shear displacement and mark the transition from theinitial post-peak decrease in shear resistance to the resid-ual shear resistance.

• Stage 5 At this point, a combination of tensile and shearcracks have extended across the base of the asperity. Asa result, further shear displacement can proceed alongan effectively flattened interface. Thus, the contributionof the asperity to the overall shear resistance has beenreduced to the frictional resistance between the brokenpieces of the original asperity (gouge) and the intact frac-ture walls (i.e., residual shear strength).

13

Fig. 16: (a) Evolution of asperity stress state; (b) Damage patterns observed via µCT imaging in ‘sawtooth’ and natural discontinuity geometries; and (c)simulated stress distribution and damage evolution.

4. Concluding remarks

This thesis summary has described a two-facetted method-ology to investigate asperity degradation and discontinuitymorphology for discontinuities subjected to direct shear test-ing. A new experimental methodology, including the de-sign, fabrication, and shear testing of discontinuity replicastogether with µCT imagery was used to quantitatively charac-terize the changes in specimens. Afterwards, the modelling ofthe direct tests was undertaken using FDEM, as implementedin the open-source Y-Geo code. The combination of experi-mental and numerical results were used to better understandthe asperity degradation process and establish the magnitudeof changes to the discontinuity geometry resulting from sheardisplacement under constant normal load.

This study marked the first use of µCT imaging to char-acterize discontinuity morphology related to shear displace-ment. These images provided the first glimpses of asperitydamage within the interior of a direct shear specimen with-out separating the joint walls or destroying the sample via se-rial sectioning. The images clearly illustrated relative differ-ence in damage and changes in aperture as a result of varying

constant normal loads and increasing shear displacement. Inaddition to the novel application of µCT imaging, this studymarked the first comprehensive attempt at simulating discon-tinuity shearing using a FDEM code. This modelling ap-proach was shown to reasonably reproduce the experimentallyobserved progressive degradation of asperities in direct sheartests in terms of the shear resistance and dilation as a functionof shear displacement. However, the shear stiffness and grad-ual post-peak decrease in shear resistance displayed notabledifferences compared to the experimental results due to thelimitation of considering 2D cross-sections instead of 3D dis-continuity surfaces. Nevertheless, with the limitation of themodelling well-understood, it can now begin to be applied tolarger-scale problems where physical testing is impractical.

Overall, this work has led to an improvement in the un-derstanding of the link between the asperity damage and thecorresponding stress conditions. In particular, the relation-ship between the observed asperity fracture pattern and themagnitude and orientation of the principal stresses was es-tablished. Tensile failure was identified as the dominant fail-ure mechanism with shear failures playing a secondary roleand, based on these findings, a new generalized conceptual

14

Fig. 17: Five stage conceptual model of asperity degradation and its relation-ship with the different stages of a direct shear test.

model for the breakdown of asperities was proposed. Thismodel clarifies the mechanisms of progressive asperity degra-dation which were ambiguous prior to the current study. Theimproved understanding gained through this study will beof value to several areas of rock engineering and hydroge-ology, including those concerned with preventing shear dis-placement (e.g., excavation, slope, and dam stability analyses)and those concerned with changes in hydraulic transmissivityresulting from shear displacement (e.g., long-term radioactivewaste repositories and reservoir stimulation by hydraulic frac-turing).

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