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CHARACTERIZATION OF IMPACT-INDUCED DAMAGE
OF JOINTED ROCK MASSES
by
Shabnam Aziznejad
A thesis submitted in conformity with the requirements for the degree of Master of Applied Science
Department of Civil Engineering University of Toronto
© Copyright by Shabnam Aziznejad, 2015
ii
Characterization of Impact-induced Damage of Jointed Rock
Masses
Shabnam Aziznejad
Master of Applied Science
Department of Civil Engineering
University of Toronto
2015
Abstract
The dynamic response of rock under impact loading has become of particular interest for several
mining and civil engineering applications. This thesis uses an advanced numerical method to
address the response of jointed rock masses subjected to impact loading. The applied
methodology includes the determination of static and dynamic mechanical properties of an intact
rock unit by conducting a series of laboratory tests. A distinct element code, 2D Particle Flow
Code, was used to generate a bonded particle model to simulate the intact rock properties. This
was followed by simulation of large-scale jointed rock mass samples by incorporating joint
networks into the calibrated bonded particle model. Finally, the impact-induced damage inflicted
by a rigid projectile particle on the jointed rock mass samples was determined. A parametric
analysis was performed to investigate the influence of joint characteristics, impact condition, and
projectile particle properties on the damage inflicted by impact on the simulated jointed rock
mass samples.
iii
Acknowledgments
This thesis would not have been possible without the guidance and the help of those who
contributed in a significant way to my research progress. First, I would like to express my deep
gratitude to my supervisor, Professor Kamran Esmaieli for his patience, enthusiasm, and the
continuous support of my research. His guidance helped me during the time of research and
writing of this thesis. My sincere thanks go to Professor John Hadjigeorgiou (my co-supervisor)
for his constructive comments and suggestions.
I would like to appreciate the contribution of Mr. Denis Labrie from CANMETmining
laboratory, National Resource of Canada, Ottawa. Special thanks to Matthew Purvance and Jim
Hazzard from Itasca, for their help on modeling with the Particle Flow Code.
I would like to thank my friends, Amin Jafari who always helped me with his valuable
suggestions and Ardavan Amirchoupani for his assistance in editing this thesis.
I should also acknowledge the financial support of this project provided by Connaught Early
Research Award, the University of Toronto. Access to the latest version of the PFC code was
made possible by a grant from Itasca.
Finally but foremost, I would like to appreciate my family for their endless love and
unconditional support. Without them, it was impossible for me to reach this point of my life.
November 2014
Toronto, Canada
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Table of Contents
Abstract ........................................................................................................................................... ii
Acknowledgments .......................................................................................................................... iii
List of Tables ................................................................................................................................ vii
List of Figures .............................................................................................................................. viii
List of Abbreviations .................................................................................................................... xii
Chapter 1: Introduction ................................................................................................................... 1
1.1 Background ......................................................................................................................... 1
1.2 Problem Definition .............................................................................................................. 2
1.3 Objectives ........................................................................................................................... 3
1.4 Methodology ....................................................................................................................... 3
1.5 Thesis Structure .................................................................................................................. 4
Chapter 2: Impact-induced Damage of Jointed Rock Masses ........................................................ 5
2.1 Introduction ......................................................................................................................... 5
2.2 Examples of Impact induced damage of jointed rock masses ............................................ 5
2.2.1 Ore pass wall wear in underground mines .............................................................. 5
2.2.2 Rockfall ................................................................................................................... 9
2.2.3 Drilling and Blasting ............................................................................................. 10
2.3 Dynamic Response of Rock .............................................................................................. 10
2.3.1 Experimental Techniques for Determination of the Mechanical Properties of
Rock under Dynamic Loading .............................................................................. 11
2.4 Characterization of Impact induced Damage of Rock ...................................................... 12
2.4.1 Experimental studies ............................................................................................. 14
2.4.2 Analytical Methods ............................................................................................... 18
2.4.3 Numerical Modeling ............................................................................................. 19
v
2.5 Summary ........................................................................................................................... 23
Chapter 3: Simulation of Static and Dynamic Properties of an Intact Rock using Bonded
Particle Model .......................................................................................................................... 25
3.1 Introduction ....................................................................................................................... 25
3.2 Experimental Tests ............................................................................................................ 25
3.2.1 Uniaxial Compression Test ................................................................................... 26
3.2.2 Brazilian Test ........................................................................................................ 27
3.2.3 Drop Test .............................................................................................................. 27
3.3 Numerical Modeling ......................................................................................................... 30
3.3.1 Simulation of the Intact Rock using a Bonded Particle Model ............................. 30
3.3.2 Advances in Bonded Particle Model (BPM) ........................................................ 30
3.3.3 Model Generation ................................................................................................. 35
3.3.4 Model calibration with static and dynamic properties of intact rock .................... 36
3.4 Summary ........................................................................................................................... 43
Chapter 4: Impact-induced Damage of Jointed Rock Masses ...................................................... 44
4.1 Introduction ....................................................................................................................... 44
4.2 Simulation of Jointed Rock Mass ..................................................................................... 45
4.3 Simulation of Impact Test on Jointed Rock Mass Samples .............................................. 46
4.3.1 Impact Force on the Rock Mass Samples ............................................................. 47
4.3.2 Impact-induced Stresses on the Rock Mass Samples ........................................... 47
4.3.3 Impact-induced Damage of the Rock Mass Samples ........................................... 48
4.3.4 Energy Transformations During the Impact ......................................................... 48
4.4 Effect of Joint Characteristics on Impact-Induced Damage of Rock Masses ................... 49
4.4.1 The Effect of Joint Intensity ................................................................................. 50
4.4.2 The Effect of Joint Orientation ............................................................................. 56
4.5 Effect of Impact Condition on Impact-Induced Damage of Rock Masses ....................... 61
vi
4.5.1 The Effect of Impact Velocity .............................................................................. 62
4.5.2 The Effect of Impact Angle .................................................................................. 66
4.6 Effect of Projectile Particle Properties on Impact-Induced Damage of Rock Masses...... 71
4.6.1 The Effect of Projectile Particle Density .............................................................. 71
4.7 Summary ........................................................................................................................... 74
Chapter 5: Conclusions and Future Work ..................................................................................... 76
5.1 Introduction ....................................................................................................................... 76
5.2 Summary of the research work ......................................................................................... 76
5.3 Conclusions ....................................................................................................................... 77
5.4 Limitation of the employed methodology ......................................................................... 78
5.5 Future work ....................................................................................................................... 78
References ..................................................................................................................................... 80
Appendix A: Laboratory Test Results .......................................................................................... 76
Appendix B: Bonded Particle Model Genesis Procedure ............................................................. 93
vii
List of Tables
Table 3.1: Laboratory Test Results for Meta-sandstone Rock Samples ...................................... 27
Table 3.2: Developments of Bonded Particle Model .................................................................... 30
Table 3.3: Clump properties for simulating the drop plate ........................................................... 38
Table 3.4: Calibration factors for the contact between the drop plate and the test specimen……39
Table 3.5: Micromechanical properties of the calibrated bonded particle model (BPM) ........... 40
Table 3.6: Comparison between experimental test results and the PFC2D (BPM) simulation .... 41
Table 4.1: Micro-mechanical properties of smooth-joint fractures within SRM samples ............ 45
viii
List of Figures
Figure 2.1: Damage and wear zones in an ore pass (after Esmaieli and Hadjigeorgiou, 2009). .... 7
Figure 2.2: Longitudinal section of the CMS results for an ore pass. ............................................ 8
Figure 2.3: Schematic diagram of strain rate regimes (in reciprocal seconds) and the techniques
that have been developed for obtaining them (Field and Walley, 2004) ...................................... 12
Figure 2.4: General picture of crack system under impact in the longitudinal section (Linqvist
1994). ............................................................................................................................................ 13
Figure 2.5: Schematic diagram of one-stage gas gun experiment device (Gao et al., 2010) ....... 15
Figure 2.6: Experimental setup for solid particle impact experiment conducted by Momber
(2003) ............................................................................................................................................ 16
Figure 2.7: Schematic diagram of edge-on impact experiment: a) High speed camera observation.
b) sarcophagus configuration. ....................................................................................................... 17
Figure 3.1: SEM examination of the samples showing sub rounded quartz and albite grains
displaying a preferred orientation (red line) set in a matrix of fine grained clinochlore and
muscovite ...................................................................................................................................... 26
Figure 3.2: (a) Measuring compressive strength and static tensile strength of rock specimens
using the MTS 815 load frame for quasi-static tests; (b) Dynatup 9210 load frame for dynamic
impact tests; (c) Specimen set-up ready for dynamic testing; (d) Failed specimen after dynamic
test. ................................................................................................................................................ 29
Figure 3.3: Illustrations of contact and parallel bond models in PFC (Potyondy and Cundall
,2004) ............................................................................................................................................ 32
Figure3.4: Particle rotation mechanisms in clustered (right) and clumped particles (left) (Choe et
al., 2007) ....................................................................................................................................... 33
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Figure 3.5: GBM consisting of grains (bonded disks) and interfaces (smooth-joint contacts)
(Potyondy, 2010) ........................................................................................................................... 34
Figure 3.6: 2D flat-joint contact (left) and flat-jointed material with effective surface of one grain
highlighted (right) (Potyondy, 2012a). ......................................................................................... 35
Figure 3.7: Simulation of an intact rock specimen using BPM in PFC2D and effective interface
geometry of flat-joint contact model between balls. ..................................................................... 36
Figure 3.8: (a) Stress-Strain response from a compression test; (b) Micro crack distribution
within the specimen. ..................................................................................................................... 38
Figure 3.9: Micro crack distribution within a BPM during a simulated static Brazilian test. ...... 38
Figure 3.10: Drop test simulation with PFC2D (left) versus a real laboratory test (right). .......... 39
Figure 3.11: Behavior and components of linear contact model. ................................................. 41
Figure 3.12: Micro cracks developed within a test specimen according to a BPM drop test. ...... 41
Figure 4.1: Smooth joint geometry (Diego Mas Ivars et al., 2011) .............................................. 45
Figure 4.2: Measuring circles of different radii within the rock mass samples to compute the
impact induced stresses ................................................................................................................. 48
Figure 4.3: Fractured rock mass samples with different fracture intensities. ............................... 51
Figure 4.4: Impact test on the fractured rock mass sample. .......................................................... 51
Figure 4.5: Impact-induced damage within rock mass samples with different fracture intensities:
a) P21 = 0 m-1; b) P21=1 m-1; c) P21 = 2 m-1; d) P21 = 3 m-1. Tension and shear cracks are indicated
in red and black, respectively. ....................................................................................................... 53
Figure 4.6: Influence of rock mass fracture intensity on the number of impact-induced micro
cracks. ........................................................................................................................................... 54
x
Figure 4.7: Impact induced stresses at different distances from the impact point within the rock
mass samples with different fracture intensity .............................................................................. 54
Figure 4.8: The effect of fracture intensity on the coefficient of restitution……………………..55
Figure 4.9: Fractured rock mass samples with different fracture orientation……………………56
Figure 4.10: Impact-induced damage within rock mass samples with different fracture
orientation: a) dip = 0º; b) dip =45º; c) dip = 90º; Tension and shear cracks are indicated in red
and black, respectively. ................................................................................................................. 58
Figure 4.11: Influence of fracture orientation on the number of impact-induced micro cracks…58
Figure 4.12: Impact induced stresses at different distances from the impact point within the rock
samples with different fracture orientations. ................................................................................ 59
Figure 4.13: The effect of fracture orientation on the coefficient of restitution ........................... 60
Figure 4.14: Impact test on the fractured rock mass sample. ........................................................ 62
Figure 4.15: Impact-induced damage within rock mass samples with different impact velocity: a)
V = 5 m/s; b) V =10 m/s; c) V = 15 m/s; Tension and shear cracks are indicated in red and black,
respectively. .................................................................................................................................. 63
Figure 4.16: Influence of impact velocity on the number of impact-induced micro cracks. ........ 64
Figure 4.18: The effect of impact velocity on the acting force on the rock samples .................... 64
Figure 4.17: Impact induced stresses at different distances from the impact point within the rock
samples with different impact velocities. ...................................................................................... 65
Figure 4.19: The effect of impact velocity on the coefficient of restitution ................................. 65
Figure 4.20: Impact test on the fractured rock mass sample with different angles. ..................... 66
xi
Figure 4.21: Impact-induced damage within rock mass samples with different impact angle: a) α
= 30º; b) α = 60º; c) α = 90º; Tension and shear cracks are indicated in red and black,
respectively. .................................................................................................................................. 67
Figure 4.22: Influence of impact angle on the number of impact-induced micro cracks. ............ 68
Figure 4.23: The normal and shear impact force acting on the rock mass samples for impact tests
....................................................................................................................................................... 69
Figure 4.24: Impact induced stresses at different distances from the impact point within the rock
samples with different fracture orientation ................................................................................... 70
Figure 4.25: The effect of impact angle on the coefficient of restitution ..................................... 70
Figure 4.26: Impact test on the fractured rock mass sample. ........................................................ 71
Figure 4.27: Impact-induced damage within rock mass samples with different projectile particle
density: a) d = 2100 Kg/m3; b) d =2700 Kg/m3; c) d = 3000 Kg/m3; Tension and shear cracks are
indicated in red and black, respectively. ....................................................................................... 72
Figure 4.28: Influence of projectile particle density on the number of impact-induced micro
cracks. ........................................................................................................................................... 73
Figure 4.29: The normal impact force acting on the rock mass samples for impact tests ............ 73
Figure 4.30: Impact induced stresses at different distances from the impact point within the rock
samples with different projectile particle densities ....................................................................... 74
xii
List of Abbreviations
ASTM
BPM
CBPM
ClmPM
ClsPM
CMS
DEM
GBM
PFC
SEM
SJM
SRM
UCS
American Society for Testing and Materials
Bonded Particle Model
Conventional Bonded Particle Model
Clump Particle Model
Cluster Particle Model
Cavity Monitoring Survey
Discrete Element Model
Grain Based Model
Particle Flow Code
Scanning Electron Microscopy
Smooth Joint Model
Synthetic Rock Mass
Uniaxial Compression Strength
1
Chapter 1 Introduction
1.1 Background
Rock materials are subjected to dynamic loading in many civil and mining applications. They
often have to withstand not only the static loads but also the impact loads due to explosions or
collisions with other objects. Hence, dynamic response of rocks under impact loading has
become of particular interest in several mining and civil engineering applications including the
followings:
In underground mines, ore pass systems are commonly used to transfer ore or waste from
one mining level to a lower level using gravity. The material flow in an ore pass can
cause wear along the ore pass walls by both impact-induced damage and frictional
abrasion. A better understanding of the impact-induced damage mechanism is therefore
essential for minimizing the wear of ore pass walls, mitigating the economic
consequences, and extending the service life of an ore pass.
In a rockfall event, rock blocks fall along a natural or engineered slope. Rockfall events
can cause severe safety hazards (i.e. loss of life) and economic losses (i.e. damage to
infrastructures, open pit mines, etc.). The process of rock block fragmentation upon
impact in rockfalls is the most complicated and poorly understood aspect of rockfall
analysis. Moreover, the interaction between a falling rock block and a slope surface still
requires further investigation.
In most mining and civil engineering projects drilling and blasting are used for rock
excavation. In a percussive drilling process, the drilling bit continuously hits the rock
material in order to destruct the rock and drill a hole. Similarly, in blasting rock material
is subjected to shock due to explosives. In both cases the rock fracture mechanism and
fragmentation process occurs under impact loading conditions. Investigation of rock
damage due to dynamic impact loading allows for a better understanding of energy
2
dissipation in drilling and blasting process and consequently a more reliable prediction of
rock fragmentation.
The behavior of rocks under impact loading conditions is a topic of great interest and importance
given its potentially wide utilization in geomechanics related issues.
1.2 Problem Definition
Modeling the response of rock material is a challenging area of research and it is more complex
under dynamic loading condition due to the transient nature of this loading. Although many
studies on rock material response under static conditions can be found, due to the complexity of
dynamic problems, fewer studies have been conducted on the behavior of rock under dynamic
loading. In addition, presence of pre-existing discontinuities in rock such as joints, bedding
planes, foliations, etc, could significantly influence the response of rock materials under dynamic
loads.
Several researchers have successfully used experimental methods to quantify the response of
rock materials to impact loading, (Camacho and Ortiz, 1996; Xia and Ahrens, 2001; Momber,
2003; Grange et al., 2008; Gao et al., 2010; Cao et al., 2011). However, since dynamic impact
laboratory tests are costly and inherently complex, obtaining such data becomes both expensive
and difficult. In recent years, some researchers have focused on theoretical models for the
evolution of rock damage under impact loading, (Taylor et al., 1986; Ahrens and Rubin, 1993;
Hu and Li, 2006). However, an issue in these analytical models is that there are too many
parameters involved and it is difficult to ascertain them. In addition, only a few of these
investigations deals with practical applications. Some researchers have employed the numerical
modeling tools in order to determine the behavior of rock materials under impact loading,
(Taylor et al., 1986; Beus et al., 1999; Nazeri et al., 2002; Wang and Tonon, 2010; Gao et al.
2010; Esmaieli and Hadjigeorgiou, 2011). Despite these extensive studies that have been
conducted on rock materials, limited investigations have been carried out to quantify the impact-
induced damage of jointed rock masses.
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This thesis uses a numerical method to address the response of jointed rock masses that are
subjected to impact loading. The study allows to inherently link micro- and macro-scale rock
fracture mechanisms with important engineering applications in mining and civil engineering.
1.3 Objectives
The main objective of this thesis is to better understand the mechanisms of impact-induced
damage of jointed rock masses. The secondary objectives are:
To simulate the static and dynamic properties of an intact rock using a numerical method
To quantify impact-induced damage inflicted on a jointed rock mass using a numerical
approach
To investigate the influence of joint characteristics, impact condition, and projectile
particle properties on the damage inflicted by impact on a jointed rock mass
1.4 Methodology
The first step of the methodology for this thesis was the determination of static and dynamic
mechanical properties of an intact rock unit by conducting a series of laboratory tests on meta-
sandstone rock samples retrieved from a quarry in Nova Scotia. This was followed by simulating
both static and dynamic properties of the intact rock unit. A distinct element code, 2D Particle
Flow Code (PFC2D), was used to generate a bonded particle model to simulate the static
properties such as uniaxial compression strength (UCS), elastic modulus, Poisson’s ratio, and
tensile strength as well as the dynamic tensile strength of the intact rock. The calibrated
numerical model was then used to develop large-scale jointed rock mass samples by
incorporating joint networks into the bonded particle model. Finally, the impact-induced damage
inflicted by a rigid projectile particle on the jointed rock mass samples was determined. A
parametric analysis was performed to investigate the influence of joint characteristics (intensity
and orientation), impact condition (velocity and angle), and projectile particle property (density)
on the response of the jointed rock mass samples subjected to the impact loading.
4
1.5 Thesis Structure
In this thesis, an investigation into mechanics of impact-induced damage in jointed rock masses
was undertaken. A brief background on the nature of this work was discussed in this chapter.
Chapter 2 provides a comprehensive literature review on the dynamic behavior of rock under
impact loading. In this chapter the experimental studies and the analytical and numerical models
developed for description and quantification of the impact-induced damage of rock material are
reviewed.
Chapter 3 illustrates the intact rock simulation as a bonded particle model. To determine the
static and dynamic properties of intact rock, a series of experimental tests were conducted. Using
the experimental results, a bonded particle model was generated and calibrated.
Chapter 4 presents the results of a series of numerical impact test results which was used to
investigate the damage induced by rigid particle impact on the jointed rock mass. The jointed
rock mass samples were generated by incorporating fractures into the calibrated large-scale
bonded particle model. Finally, the effects of joint characteristics, impact condition, and
projectile particle properties on the damage induced by projectile particle impact on the rock
mass were investigated.
Finally, chapter 5 provides a summary and conclusions of this research project as well as
recommendations for future work.
5
Chapter 2 Impact-induced Damage of Jointed Rock Masses
2.1 Introduction
In the field of rock mechanics, the behavior of rock materials under impact loading is of
particular concern and interest in several applications. This chapter addresses issues concerning
impact-induced damage of rocks. Degradation of ore pass walls due to material flow impact,
fragmentation of rock blocks upon impact with rock slope surfaces in rockfall events, and
impact-induced rock fragmentation through straight impact or by explosive in drilling and
blasting are discussed as part of this chapter to emphasize the importance of this phenomenon in
engineering applications.
A brief discussion follows on the experimental techniques to determine the dynamic properties
of rock materials. Finally, the methods for characterization of damage inflicted on rock material
upon impact loading are introduced and different approaches such as experimental studies,
analytical methods, and numerical models are briefly explained.
2.2 Examples of Impact induced damage of jointed rock masses
Rock masses are subjected to impact loading in several engineering applications. In the
following sections, a review has been conducted on some practical examples where the response
of rock mass under impact loading is of importance and concern.
2.2.1 Ore pass wall wear in underground mines
Ore passes are a series of vertical or steeply inclined underground excavations that are
commonly used in underground mines to transfer ore or waste from one mining level to a lower
level using gravity. They are considered an important element of material handling system.
Efficient operation of ore passes in an underground mine leads to a more reliable material
handling system and a more profitable mine operation. Ore pass systems frequently encounter
problems with the degradation of their structural integrity, resulting in gradual failure or sudden
instability in the structural integrity of the ore pass. The degradation of ore pass walls results in
6
the expansion of its original dimensions. This expansion can influence the stability of adjacent
openings and the reliability of the material handling system. It can also result in operational
deficiencies of the ore pass system that can lead to safety hazards in mining operations and
substantial economic consequences in forms of diminished mine production while imposing high
rehabilitation costs, (Lessard and Hadjigeorgiou, 2007).
Ore pass wall degradation can be the consequence of simultaneous interaction of various
mechanisms including changes in the stress state around ore pass walls, structural failure due to
the presence of natural discontinuities in the rock mass around the ore pass, and wear due to
material flow.
Dumping broken ore and waste materials into an ore pass can cause wear along the ore pass
walls. The wear is the result of rolling, sliding and colliding of fragmented rock boulders with
the ore pass walls under the gravity force. Wear of an ore pass wall can be caused by the impact-
induced damage and by abrasion, (Goodwill, 1999). Figure 2.1 shows the potential development
of wear and/or impact damage zones along ore pass walls. The fragmented rock boulders can be
dumped directly into an ore pass or be funnelled to the ore pass via a finger raise.
7
Figure 2.1: Damage and wear zones in an ore pass (after, Esmaieli and Hadjigeorgiou, 2009).
Morrison and Kazakidis (1995) carried out a qualitative assessment of the ore pass wall
degradation in mines located at the Sudbury Basin in Canada. In this study, they concluded that
frictional abrasion and wear due to impact loading caused by material flow were the two most
important mechanisms which contributed to the ore pass wall degradation. Based on the survey
of over 10 underground mines in Canada, Hadjigeorgiou et al. (2005) reported that more than
40% of the ore pass sections surveyed were showing some signs of ore pass wall degradation
due to wear caused by impact loads and frictional abrasion. Cavity monitoring surveys (CMS)
are commonly used to measure and evaluate the localization and extent of the degradation zones
along ore passes. Figure 2.2 presents a longitudinal section of the CMS results for an ore pass in
a Canadian underground mine. The results indicate a significant expansion of the ore pass wall
after more than four million tons of rock was dumped into the ore pass via the finger raise.
Specifically, it was observed that the ore pass wall facing the finger raise expanded up to about
four to six times its original dimensions mainly due to impact loads on the ore pass wall.
8
Figure 2.2:Longitudinal section of the CMS results for an ore pass (after Esmaieli and
Hadjigeorgiou, 2011).
In another study, Hadjigeorgiou and Mercier-Langevin (2008) demonstrated that ore pass wall
impact from material flow is one of the important factors that influence the longevity of an ore
pass. The ore pass wall damage attributed to impact loading is most often localized at the
intersection of finger raises with the ore pass.
Experience shows that presence of discontinuities in the rock mass along an ore pass wall can
considerably increase the rate of ore pass wall wear. In addition, the geometrical characteristics
of discontinuities have an influence on the erosion and abrasion rate of the rock mass along the
ore pass wall, (Joughin and Stacey, 2005), (Hadjigeorgiou et al., 2005). Despite these
experimental and observational insights, limited studies have been conducted to quantify the
impact-induced damage of jointed rock masses. Most recently, Esmaieli and Hadjigeorgiou
(2014) conducted a series of numerical experiments to investigate the influence of the rock mass
foliation angle on impact-induced damage of ore pass walls. The investigation demonstrated that
the presence of foliation planes results in greater damage zone in the jointed rock mass. It was
9
also shown that the extent of impact-induced damage to the ore pass walls, with foliation planes
oriented semi-parallel to the wall, is greater than those with foliation planes oriented
perpendicular to ore pass walls.
To predict how long an ore pass system will be functional, it is often necessary to quantify the
results of the interaction between impact load and rock mass on ore pass wall degradation. A
better understanding of the impact-induced damage mechanism of jointed rock masses is
important for minimizing the wear of ore pass wall, mitigating the economic consequences, and
taking action to extend the life of an ore pass.
2.2.2 Rockfall
A rockfall event involves the fall of rock blocks or fragments along a natural or engineered
slope. In open pit mines, rockfalls are generally associated with geological features, weathering
of the rock surfaces, blast-induced fracturing, and machinery activity on top of crests and along
the roads at the bottom of highwalls, (Giacomini et al., 2012).
Rockfalls could pose a severe safety hazard and cause significant damage to infrastructure in
mines. Such events could have major economic consequences due to the need to maintain or
rebuild of infrastructures and temporary suspension of production for safety reasons. Therefore,
the safety implications associated with rockfall events need to be managed in surface mining
operations. To ensure this, there is a real need to improve our understanding of the complexity of
the interaction processes between the falling rock blocks and the slope.
Many researchers developed field experiments in order to gather in situ parameters describing
the impact phenomenon of rockfalls, (Giani et al., 2004). To date, research efforts have been
focused on in situ rockfall tests, barrier tests, and the development of analytical and numerical
models to evaluate the trajectories of detached blocks upon impact to the slope surface,
(Giacomini et al., 2009).
Rock fragmentation is frequently observed upon impact of a rockfall event. The physical process
of rock fragmentation during rockfall is complex because it consists of impact-induced stress
waves that propagate, generate thermal energy, and create plastic zones. Presence of
10
discontinuities in the falling rock boulders, high impacting energy and rigidity of ground slope
can facilitate rock fragmentation, (Wang, 2009). Given that only a small number of studies
regarding the fragmentation of rock blocks along their path have been carried out, it could be
argued that this aspect of rockfall analysis has been poorly. Most of the previous studies have
focused their efforts on understanding the effects of the strain rate on the rock fragmentation
phenomenon via energy considerations, (Giacomini et al., 2009). However, the impact-induced
rock fragmentation process in rockfall analysis cannot be well predicted via energy
considerations alone because the process is not only controlled by the energy dissipation, but
also related to other factors such as impact velocity, incidence angle, pre-existing joints, and
ground stiffness, (Wang and Tonon, 2010).
2.2.3 Drilling and Blasting
In most mining and civil engineering projects drilling and blasting is used for rock excavation.
Different drilling methods are available based on the mode of energy application to the target
rock leading to the fragmentation of the rock. The most successful method for practical
applications is drilling by mechanical means in which a heavy bit is raised and lowered onto the
rock. Most of the research work has focused on the fracture pattern that can form on the rock due
to the static or quasi-static indentations forces. However, in actual percussive drilling processes,
the drilling bit impacts the rock at 5-15 m/s velocity, (Saksala, 2010). For this reason, the
fracture and fragmentation process in rocks should also be studied under dynamic loading.
Limited studies have been conducted to understand the interaction between pre-existing
discontinuities and the fractures induced by impact of drill bits.
Additionally, impact-induced rock fragmentation through straight impact or by explosive is of
particular interest in blast theory research. In order to investigate the rock fragment distribution
after blasting, it is very important to understand the rock mass response under impact loading.
2.3 Dynamic Response of Rock
Dynamic loads are presented in the form of high amplitude stress waves in short duration. The
transient nature of dynamic loading makes it very different and more challenging to understand
than the static ones. Despite quasi-static loading conditions, the mechanical behavior of materials
11
becomes more complex under dynamic loading conditions. Under this condition, two basic
phenomena occur: 1) the change in the mechanical behavior of material with varying the strain
rate, and 2) evolution and propagation of shock waves, (Hiermaier, 2013).
The mechanical properties or rock are altered under different strain rates varying from creep (ε
=10-8 s-1) to shock (ε =108). As a result of the rate dependent property of rock materials, a
significant increase in mechanical properties including tensile strength, compression strength,
failure strain and fracture energy can be observed by increasing the strain rates. The reason
behind this phenomenon can be found in the differences in fracture process. The effects of inertia
and stress wave propagation make the dynamic fracture process to be much more complicated
than the static ones. At low strain rates, the fracture process starts from existing micro cracks and
discontinuities and develops along the path which requires the least energy. Therefore, limited
extension of micro-cracks can be found in high strength areas due to low overall stress and
relaxation of such regions. However, at high strain rates, the existing cracks can develop along
high resistance areas because of the large amount of energy dissipation which occurs in a very
short time. Since relaxation cannot occur in this short time, extensive micro-cracks are found as a
result of the rapid increase in tensile stresses, (Cadoni, 2013).
The interaction between the stress waves and rock materials affects stress wave attenuation and
rock failure. In dynamic loading conditions, the effects of inertia become more important. This is
because high amplitude stress waves increase the strength of rock materials leading to high
density fracturing which results in the failure of the material with more fractures.
2.3.1 Experimental Techniques for Determination of the Mechanical
Properties of Rock under Dynamic Loading
There are a variety of techniques available to obtain the dynamic mechanical properties of rock
under a wide range of strain rates (i.e. different loading condition) including drop weight
machines, Split Hopkinson pressure bar, Taylor impact, and shock loading by plate impact,
(Field and Walley, 2004). High-speed photography and optical techniques are also recently used
to study ballistic impact, (Field and Walley, 2004).
12
Conventional mechanical testing machines cover the low strain rate range of up to around 10 s-1.
Drop weight machines cover the strain rate range of 10-103 s-1. For testing rock material under
high strain rate (103 ~ 104 s-1), Split Hopkinson pressure bar is an ideal choice and for higher
strain rates, other techniques can be used, (Field and Walley, 2004). Figure 2.3 shows a
schematic diagram of the developed techniques for testing rock material under different strain
rate regimes.
Figure 2.3: Schematic diagram of strain rate regimes (in reciprocal seconds) and the techniques
that have been developed for obtaining them (after, Field and Walley, 2004).
One important point in Figure 2.3 is the transition from 1D stress to 1D strain which occurs by
increasing the strain rate. This transition is due to inertial confinement of the material. The strain
rate at which this transition occurs depends on the density and the size of the specimen.
Therefore, larger specimen with higher density results in lower transitional strain rates, (Field
and Walley, 2004).
2.4 Characterization of Impact induced Damage of Rock
When a projectile object impacts a rock material, a complicated stress field is produced in the
target rock depending on the material’s properties and impact conditions. In such a scenario,
tensile stresses develop at the contact point between the projectile and rock and propagate
radially. Furthermore, just below the impact point, zones of tension, shear, and hydrostatic stress
13
are developed and the material may fail by any of the aforementioned mechanisms.. However, it
should be mentioned that at a microstructure scale, the damage mechanism occurs in tensile
mode even in the compression loading condition.
The collision of a projectile object with a rock initially leads to the closure of pre-existing flaws.
This is followed by elastic deformation of the rock in the area of impact, followed by the ejection
of small rock particles from the rock surface, creating a crater zone. Underneath the crater zone,
a significant portion of impact energy is utilized for crushing the rock material. Beyond the
crushed zone, the rest of impact energy can create cracks of different forms and lengths,
including large radial, side, and median cracks, (Linqvist, 1994). Figure 2.4 schematically
illustrates the crack system and the impact-induced damage zones inflicted on a rock surface in a
longitudinal section.
Figure 2.4: General picture of crack system under impact in the longitudinal section (after,
Linqvist 1994).
Direct estimation of impact-induced damage involves measurement of actual crack distribution.
Conventional damage mechanics defines damage as only the micro-cracked areas of the material
and ignores the crater and crushed regions. Damage is therefore defined as the surface area of the
micro-cracks in the material to the surface area for the representative volume element at a point.
Damage varies between 0 for undamaged material and 1 for ruptured material, (Lemaitre and
Desmorat, 2005). This approach gives a low value compared to the actual damage.
14
Indirect measurement of damage includes quantification of change in the mechanical properties
of rock like the change in wave velocities, elastic modulus, micro-hardness, electrical resistance,
etc. A common approach for damage calculation is through measuring the changes in the wave
velocities in the target sample after impact, (Pellet and Fabre 2007). However, this technique
shows only an overall picture of damage. Measurement of the actual cracked volume using X-ray
Computed Tomography scanning and impregnation techniques enables better characterization of
the impact-induced damage. A more realistic damage description need to distinguish the various
aspects of impact-induced damage including cratering, crushing, and generation of crack
network.
The inelastic brittle response of rock under dynamic loading is due to stress-induced micro-
cracks. The growth and interaction of the micro-cracks makes portions of the rock volume
unable to carry load and decrease its stiffness. This response is known as strain-softening
behavior. This continuous degradation of the material moduli during the loading process reflects
the damage accumulation. Hence, the mechanics of micro-cracks is an essential part of the
understanding of the dynamic damage response of rock. At high or rapid change of stresses
induced by dynamic loading, the damage process depends on the dynamic behavior of fractures
and micro-crack propagations lead to dynamic damage evolution, (Keita et al. 2013).
2.4.1 Experimental studies
Many researchers have carried out dynamic impact tests on rock in order to determine the
dynamic failure mechanism of rock. Particle impact studies, in which a metallic pellet strikes a
brittle material such as a rock plate could provide a means of identifying the failure modes the
target, (Camacho and Ortiz, 1996). Depending on the impact velocity, specimens and pellet
undergo different level of damage ranging from inelastic deformation, surface cracks, lateral and
radial fractures and resulting to significant fragmentation.
Gao et al. (2010) performed a plate impact experiment using one-stage light-gas gun technology
to study the effects of stress wave propagation and failure characteristic of limestone. A
schematic diagram of a one-stage gas gun experiment device used for such an experiment is
shown in Figure 2.5. In this experiment, the sudden release of high pressure gas causes the flyer
15
to collide on the target rock plate at a high speed and the attenuation and dispersion effects of
compression waves were studied by researchers.
Figure 2.5: Schematic diagram of one-stage gas gun experiment device (after, Gao et al., 2010).
It was observed that the impact of the projectile on the target generated strong shock waves,
which propagates with decreasing attenuation. The dynamic mechanical properties and failure
characteristics of rock sample under impact loading were studied by recovery of impacted rock.
The results of this study show that a double-wave structure exists in the impacting rock which
includes an elastic precursor wave and following a plastic main wave. The reason of the non-
linear behavior of stress wave was explained by existing inhomogeneous structure in rock. The
results also showed that the failure of rock sample had divisional characteristics corresponding to
different damage mechanisms such as head failure zone, middle tension-compression failure
zone and tail fracture failure zone.
Xia and Ahrens (2001) impacted rock samples of San Marcos gabbro by projecting aluminium
bullets with the velocity of 800-1200 m/s in the laboratory and imaged the damage structure.
They proposed a simple model to describe the damage zone depth based on the attenuation
relation for the peak shock pressure. They suggested that the depth of damage can be used as an
independent constraint for the study of the impact cratering and that the depth of the damage
zone can be used to extract information about the impact cratering process.
Momber (2003) studied the damage on four rock types (granite, limestone, rhyolite, schist) and
two cementitious materials (fine grained and coarse grained concretes) through solid particle
impact. The experimental set-up to perform the damage tests is described in the Figure 2.6.
16
Figure 2.6: Experimental setup for solid particle impact experiment conducted by Momber
(after, Momber 2003).
Projectile particles, rounded quartz with the diameter of 0.3 to 0.6 mm, were accelerated by the
air flow with the velocity calibrated through the air pressure. For single impact study, glass beads
with a diameter of 0.8 mm and velocity of 80 m/s were used. Comparative indentation tests were
also performed with tungsten carbide ball. Furthermore, in order to inspect impact sites and
visualize surface damage, an optical microscope was used. The results of this study showed that
the materials were mostly damaged by inter-granular and inter crystalline fracture. Additionally,
for the cementitious materials, the interface between cement matrix and aggregate plays the
important role. The balance between inter-granular erosion and trans-granular fracture was
expressed by an exposure time parameter.
The damage rate was also estimated based on the specimen weight before and after damage test.
It was found that the particle diameter and velocity, hardness and elastic modulus of target, and
fracture toughness are the factors that influence the damage rate through solid particle impact.
Several damage models have been developed to estimate the power exponent of these factors
such as the one proposed by Finnie and Oh (1966). The power exponents for the particle velocity
was estimated based on the experimental results and compared to the available damage models.
Edge-on impact test is another experimental method in which a cylindrical stricker hits a slab on
the edge generating an incident wave in order to observe real time crack patterns, (Grange et al.
2008). Due to the radial motion of the medium, tensile hoop stresses are induced with a high
strain rate. Two types of edge-on impact experiments are shown in Figure 2.7. In the first one, an
17
ultra-high speed camera is used to observe the kinetics of the damage process. The second one
includes sarcophagus to maintain the fragments in the place and allows for characterization of
crack density in the target
(b)
Figure 2.7: Schematic diagram of edge-on impact experiment: a) High speed camera observation.
b) Sarcophagus configuration (after, Grange et al. 2008).
Grange et al. (2008) performed edge-on impact tests on two limestones to study the crack and
fragmentation pattern induced by dynamic loading. They observed the presence of the emerging
cracks in the crushing zone for both rocks, but the fragmentation pattern after this area was
different due to the effect of microstructure. They used a probability distribution called Weibull
model to describe the microstructure and prediction of the crack density. The experiments
showed that the shape parameter of this distribution called Weibull modulus is an important
factor in the estimation of rock fragmentation. When the Weibull modulus is high, numerous
cracks can be observed.
Cao et al. (2011) conducted a series of impact experiments using ball bullet with low velocity
(10 m/s - 40 m/s) on the granite plate by the compressed gas of the Split Hopkinson pressure bar.
(a)
18
The experimental results included the data of the crushed zone and the crack length under
different dynamic impact loading. They indicated that when the local stress concentration
exceeds the material strength, the local damage occurs in forms of local cracks. Then this stress
concentration moved to the crack tip and contributed to the continuous damage development.
With the increase in the incident energy, the crushed zone and crack length increases, however
the crack growth rate was decreasing. They also investigated the effect of incident angle and
showed that the crack propagation and crushed area are not arbitrary and depend on the impact
angle. With increasing the impact angle, the crushed zone gradually increases.
Huang and Shi (2013) studied the energy absorbing ability of porous rock using Split Hopkinson
pressure bar experiments. The energy absorbing ability of material is important since it affects
the overall impact resistant. They investigated the influence of the porosity of sandstone on the
energy dissipation and the experimental results indicated that the energy dissipation increases by
increasing the porosity.
2.4.2 Analytical Methods
Many researchers have focused on the damage evolution of rock material experiencing dynamic
loading and proposed analytical models for this phenomenon.
Kachanov (1986) provided a comprehensive review of the continuum damage mechanics theory.
The conventional damage models in the literature combine the theory of fracture mechanics
which governs the behavior of individual cracks with some statistical concepts to account for the
random distribution of the micro-cracks. In fact, continuum damage models introduce internal
state variables for the distribution of micro-cracks in the material.
Taylor et al. (1986) developed a constitutive model to simulate the dynamic fracture behavior of
rock. In this model, the damage mechanism has been attributed to the micro-cracking in the rock
medium. They defined damage as the volume fraction of rock that has lost the load-carrying
capability due to multiple crack growth. The principle assumption in the model was that the
growth of the micro-cracks was treated as an internal state variable in the material rather than
treating each individual crack. Also, the continuous degradation of the material moduli which
results in a strain softening behavior of rock, was considered as the damage accumulation.
19
Ahrens and Rubin (1993) demonstrated that the damage zone could be characterized by reduced
primary and secondary wave velocities due to the initiation and growth of cracks. The amount of
damage was defined as D = 1- (CD/C0)2 where CD and C0 are S or P wave velocity of damaged
and undamaged rock respectively.
Based on the principle of energy dissipation in rock and damage mechanics, Hu and Li (2006)
proposed a dynamic cumulative damage model using the action density of energy for the rock
under repeated impact loading at the damage stage which is the first phase of dynamic
fragmentation of the brittle material. According to the principles of dynamic fracture, the energy
action density is expressed as a function of the elastic modulus and velocity of P-wave. There are
also two dynamic fracture thresholds depending on the stress pulse propagation through a rock.
If the value of the energy action density falls below the lower threshold, the whole impact energy
dissipates completely in the form of elastic waves in the rock and no damage happens in the
rock. When the value of energy action density is greater than the upper threshold, the dynamic
fracture condition is satisfied and the rock will be entirely fractured. As the energy density
approaches the upper threshold, the energy dissipation in elastic waves is 10% to 20% of the
whole loading energy. As a result, the rock will be damaged. If the rock is subjected to repeated
impact loading, the rock may be completely fractured.
2.4.3 Numerical Modeling
Experimental impact tests on large-scale rock mass samples are expensive and difficult to carry
out. On the other hand, most of the proposed damage models are empirical with very limited
applicability to large-scale rock mass. In the analytical models for intact rock, there are too many
parameters and variables which make it difficult to determine all of them. Moreover, there are
few analytical investigations that deal with applications. Numerical modelling could provide a
reasonably good understanding of the impact mechanism and its complexity by the means of
simplifications. However, it is sometimes difficult to validate the result of any numerical models
based on laboratory or field observations. This section presents a discussion on what could be
considered as an appropriate numerical model to study this problem.
The numerical models that are used for simulating dynamic behavior of rocks can be classified
into two categories: indirect and direct approach. Most indirect approaches represent damage
20
indirectly by idealizing the material as a continuum medium and utilizing the degradation
measurements in constitutive relations. On the other hand, most direct approaches consider the
material to be discontinuous as a collection of structural units or separate particles bonded
together at their contact points where damage is directly simulated by the breakage of individual
structural units or bonds.
Direct modeling is especially applicable to the type of problems such as rock fracture for which
the complex constitutive behavior arising from extensive micro- and macro-cracking is difficult
to accurately characterize in terms of a continuum formulation. The predefined complex
empirical constitutive relations are replaced with simpler particle contact logic and a rich
emergent material response can be derived. Although using the experimental impact test one can
observe the final pattern of fractures and damage, but it is also important to understand the
evolution of the fracture pattern and follow various stages including build up of stress field,
formation of crater and crushed zone, and the crack system, which is only possible via numerical
simulation.
Numerical modeling is a good approach to visualize the evolution of fracture pattern under
impact. Different mathematical models are necessary to model different stages of the impact
phenomenon. The theory of elasticity can be used to model the stress state and prediction of the
fracture initiation points. Fracture mechanics can be used to model cracking and the theory of
plasticity can be used to model the crushed zone.
Different numerical tools including finite element method, finite difference method, discrete
element method and combined finite and discrete element methods are generally used to simulate
dynamic events like impact in order to explore fracture and fragmentation of rock. These
numerical programs generally calculate the nodal force and displacement by using mass,
momentum, and energy conservation equations along with initial and boundary conditions.
Behavior of material is modeled by using Equation of State, strength model, and failure model.
21
2.4.3.1 Continuum-based Models
Continuum-based models are the indirect modeling approach that simulates damage indirectly by
idealizing the material as a continuum and utilizing the degradation measurements in constitutive
relations.
Taylor et al. (1986) implemented their developed constitutive model as discussed in section 2.4.2
into the DYNA2D, an explicit finite element code for the transient, large-deformation analysis of
two-dimensional solids. A numerical example of a blasting experiment was presented and
compared to the field observations. The numerical simulation results showed that the developed
algorithm was easy to use and the constitutive model was applicable for prediction of dynamic
rock-fracture behavior.
Gao et al. (2010) simulated the one-stage light gas gun experiment as explained in section 2.4.1
using LS-DYNA, finite element code and analyzed the dynamic response of rock under impact
loading. The simulation results were in agreement with the experimental ones and the material
model could describe the dynamic response of rock very well. The results showed that the main
reason of rock failure was the joint action of longitudinal compression wave and transverse
sparse wave.
2.4.3.2 Discontinuum-based Models
Distinct element model (DEM) is a direct modeling approach of impact-induced damage and has
been successfully applied in modeling the dynamic behavior of rocks.
The Particle Flow Code (PFC), which is a discrete element code developed by Potyondy and
Cundall (2004), has been used by several researchers to investigate impact on ore pass
components due to material flow. Beus et al. (1997) carried out an ore pass hazard assessment
through laboratory and field measurements of static and dynamic loads on ore pass chutes and
control gates. Later in 1999, Beus et al. (1999) used 2D numerical modeling (PFC2D) to
simulate the dynamic effects of material flow on ore pass chutes and control gates. Nazeri et al.
(2002) used DEM simulations to investigate the effects of ore pass configurations and ore
characteristics on the dynamic and static loads of ore pass gate assemblies. Iverson et al. (2003)
22
stated that dynamic loads induced by material flow impact can cause structural failure of control
gates, chutes, and ore pass walls.
Esmaieli and Hadjigeorgiou (2011) investigated the influence of different ore pass and finger
raise configurations and the resulting impact loads on ore pass walls using PFC2D. They
compared the results of the numerical simulations to field observations at Brunswick Mine
located in New Brunswick, Canada. The results of the analysis demonstrated that the choice of
intersection angle between ore pass and finger raise can have a significant influence on the
resulting impact loads and the location and magnitude of damage to the ore pass wall. The
investigations led to a series of recommendations on selecting appropriate configurations in order
to minimize the influence of impact loads.
Wang and Tonon (2010) developed a discrete element code which is similar to PFC in order to
simulated impact-induced rock fragmentation in rockfall analysis as discussed in section 2.2.2.
They used a simplified impact model based on the theory of vibrations for foundations on elastic
media. The theory explained how the mass spring-dashpot analog can describe the vertical and
tangential oscillations of a rigid circular disk foundation, (Richart et al. 1970). Dashpots are used
in this model to provide damping that accounts for the energy loss due to the elastic stress waves
propagating away into the ground. They investigated the effect of impact velocity, ground
condition and fracture properties on the mechanisms of rock fragmentation upon impact. In this
work, the effect of ground cratering effect on rock fragmentation upon impact was not
considered; however, it can be accounted for by introducing coefficients of restitution. The
material damping also has an effect on the fragmentation process, but was not investigated in the
analysis.
Wang and Tonon, (2010) stated that one of the challenges in modeling rockfall impact, in which
a block of rock impacts the slope ground, is to choose an appropriate model to represent the
interaction between the rock block and the slope ground including the stiffness and damping of
the ground.
In neither of these studies the effect of pre-existing discontinuities on the dynamic response of
rock has been investigated. Most recently, Esmaieli and Hadjigeorgiou (2014) conducted a series
of numerical experiments to investigate the influence of rock mass foliation angle on impact-
23
induced damage of ore pass walls. The investigation demonstrated that the presence of foliation
planes results in a greater damage zone of the jointed rock mass. It was also shown that the
extent of impact-induced damage to the ore pass walls with foliation planes oriented semi-
parallel to the wall is greater than those with foliation planes oriented perpendicular to ore pass
walls.
2.5 Summary
This chapter has reviewed the impact-induced damage mechanism of rock materials. The
behavior of rock under impact loads is a topic of great importance because of its potential
application in geomechanics related issues in mining and civil engineering. Some practical
examples where the behavior of rock under impact loading is of great interest and importance
were presented.
A review of the impact-induced damage mechanisms of rock has shown that different damage
zones including cratering, crushing and generation of crack networks can be distinguished and all
should be considered in damage quantification.
The damage of rock materials under impact loading can be evaluated by experimental, analytical
and numerical methods. Although experimental tests could be used for assessment of dynamic
behavior of small specimen of rock samples, using these tests for large-scale rock masses is
either unfeasible or too expensive. In addition, most of the developed analytical models require
too many parameters which are hard to determine.
Since field or laboratory impact tests on large-scale rock mass samples are expensive and
difficult to carry out, using numerical modelling could be a feasible solution to help to
understand mechanisms behind the impact-induced damage. Two numerical modeling
approaches, indirect and direct approaches, were also discussed along with advantages and
disadvantages of each approach.
The main objective of this thesis is to better understand the mechanisms of impact-induced
damage of jointed rock masses. This objective is achieved by using direct numerical approach.
The following chapter presents the intact rock simulation which is the first step of the
24
methodology developed in this thesis in order to study the impact induced damage of jointed
rock masses.
25
Chapter 3 Simulation of Static and Dynamic Properties of an Intact Rock
using Bonded Particle Model
3.1 Introduction
The current chapter presents the steps taken to simulate intact rock static and dynamic properties
using the bonded particle model (BPM). Intact rock laboratory data were used to simulate and
validate the numerical models. The bonded particle model suffers from the limitation of
reproducing the key characteristic of hard brittle rocks with non-linear failure envelopes. A brief
review on the evolution of the BPM since its first proposition by Potyondy and Cundall (2004) is
presented.
The following sections of this chapter includes determination of the static and dynamic
mechanical properties of an intact rock unit by using the result a series of laboratory tests on
meta-sandstone rock samples retrieved from a quarry in Nova Scotia. A distinct element code,
2D particle flow code (PFC2D), was used to generate a bonded particle model. The recently
developed contact model known as the flat-joint contact model, was successfully used to
calibrate both static (UCS, Young’s modulus, Poisson’s ratio, tensile strength) and dynamic
mechanical properties (dynamic tensile strength) of the intact rock.
3.2 Experimental Tests
Meta-sandstone rock samples from a quarry in Nova Scotia were collected and tested by the
CanmetMINING Rock Mechanics Laboratory, Natural Resources Canada, Ottawa, ON. The
composition of the rock and proportion of each mineral was determined on polished sections
through scanning electron microscopy (SEM) analysis. The electrons from the SEM devise
interact with the electrons in the sample and produce signals that contains information about the
sample composition. The collected specimens were mainly comprised of quartz and albite with
minor amounts of clinochlore, muscovite and calcite. The SEM examination of the samples
showed mostly 0.2-0.4 mm subrounded quartz grains with smaller 0.1-0.2 mm albite grains
supported in a matrix of mostly clinochlore and some muscovite (the minimum grain size of 0.1
26
mm). Figure 3.1 displays SEM examination of the samples showing the preferred orientation of
the grains due to the slight metamorphism of the rock.
Figure 3.1: SEM examination of the samples showing sub rounded quartz and albite grains
displaying a preferred orientation (red line) set in a matrix of fine grained clinochlore and
muscovite.
The collected rock samples were subjected to a series of static and dynamic tests. These include
static uniaxial compression and static and dynamic Brazilian tests. The dynamic Brazilian test
was performed using a drop-tower apparatus. The conducted laboratory tests allowed
determination of the mechanical properties of the intact rock, e.g. Uniaxial Compressive
strength, indirect static Tensile Strength, indirect dynamic Tensile Strength, Elastic Modulus and
Poisson’s Ratio.
3.2.1 Uniaxial Compression Test
For uniaxial compression tests, rock specimens were prepared and tested according to the ASTM
standard D4543 and D7012 (ASTM 2006c, 2006d, respectively). Uniaxial compression tests
were carried out on specimens with dimensions averaging 12 cm height and 5 cm diameter, using
a MTS 815 load frame for quasi-static tests. During these tests the axial load and deformation of
samples were recorded. Elastic moduli (Young’s modulus and Poisson’s ratio) were calculated
from the linear portion of the stress-strain curve, over the range of 30% to 70% of ultimate
compressive strength. Overall, 18 specimens were tested for uniaxial compressive strength.
27
3.2.2 Brazilian Test
The Brazilian test is commonly used as an indirect method for determining the tensile strength of
intact rock in the laboratory. For static Brazilian tests, meta-sandstone specimens were prepared
in accordance with ASTM standard D3967 (ASTM 2006b). Tests were carried out using the
MTS 815 load frame on specimens with dimensions averaging 5 cm diameter and 3 cm
thickness. Overall, 21 specimens were tested using the Brazilian test procedure.
3.2.3 Drop Test
To determine the dynamic tensile strength of rock specimens, dynamic drop tests were carried
out using a vertical impact load frame Dynatup Model 9210. Tests were carried out with an
impact weight of 19.35 kg, a drop height of 46 cm, for a total potential energy of 87 Joules (J) in
the system, and an impact velocity of 3 m/s. Dynamic impact tests were carried out on 44 rock
specimens.
Table 3.1 summarizes the mechanical properties of the intact rock measured in the laboratory.
Figure 3.2 shows the MTS 815 load frame for quasi-static tests and the Dynatup 9210 load
frame for dynamic impact tests. The specimen set-up for dynamic testing and Failed specimen
after the test are also shown in this figure.
28
Table 3.1: Laboratory Test Results for Meta-sandstone Rock Samples
Type of Test No. of
Specimens Properties Mean S.D.
Unconfined compression
test 18
Elastic modulus (GPa) 74.5 5.0
Poisson’s ratio 0.26 0.10
Axial strain at peak (%) 0.397 0.11
UCS (MPa) 267.0 64.0
Static Brazilian test 21 Static tensile strength (MPa) 21.5 3.0
Drop test 44
Dynamic tensile strength
(MPa) 29.8 5.0
Time to max load (ms) 0.168 0.03
Deflection (mm) 0.491 0.09
29
(a) (b)
(c) (d)
Figure 3.2: (a) Measuring compressive strength and static tensile strength of rock specimens
using the MTS 815 load frame for quasi-static tests; (b) Dynatup 9210 load frame for dynamic
impact tests; (c) Specimen set-up ready for dynamic testing; (d) Failed specimen after dynamic
test.
30
3.3 Numerical Modeling
3.3.1 Simulation of the Intact Rock using a Bonded Particle Model
Intact rock behaves like a cemented granular material with deformable and breakable complex-
shaped grains. Such a conceptual model can explain all aspects of rock mechanical behavior.
Potyondy and Cundall (2004) proposed a bonded-particle model (BPM) for the simulation of
intact rock based on distinct element method (DEM).
The BPM is based on the Particle Flow Code developed by Itasca (2014) which simulates a solid
rock as a dense packing of non-uniform-sized circular rigid particles that are bonded together at
their contact points. The procedure to generate a BPM has been discussed by Potyondy and
Cundall (2004). Newton’s laws of motion relate relative particle motion to force and moment at
each contact, which possesses finite normal and shear stiffness. Unlike the continuum numerical
models, the BPM does not impose theoretical assumptions and limitations on the constitutive
material behaviour and there is no need to develop constitutive laws. The mechanical behavior of
rock is governed by the formation and growth of micro-cracks. When a BPM is loaded,
depending on the imposed levels of stress and strain, micro cracks can initiate and develop
within the model. Cracking is explicitly simulated as a bond breakage between the rigid particles.
Micro cracks are able to form, interact, and coalesce into macroscopic fractures according to
local stress conditions.
Although the behavior of the BPM model is found to resemble that of intact rock, a particle in
BPM is not associated with a rock grain. The assembly of bonded particles is a valid
microstructural model in its own right and should not be confused with the microstructure of
rock. If it is required that the microstructure of rock be modeled in detail, then a grain-based
model will be required, (Potyondy, 2010).
3.3.2 Advances in Bonded Particle Model (BPM)
Modeling the failure of hard brittle rocks using the BPM has improved over the past ten years.
Table 3.2 presents the evolution of the BPM since its first introduction by Potyondy and Cundall
(2004). Despites the capability of BPM to model fracture initiation and propagation starting from
31
simple interaction laws, the model suffers from the limitation of reproducing the key
characteristic of hard brittle rocks with non-linear failure envelopes. The small ratio of uniaxial
compressive strength to tensile strength is a common limitation in how PFC manages intact rock
modeling. Many attempts have been done to overcome this limitation as summarized in Table
3.2. The recently developed contact model, the flat-joint contact model, seems to have
significantly improved this issue, (Potyondy, 2012).
Table 3.2: Developments of Bonded Particle Model
Year Researchers Development
2004 Potyondy and Cundall Conventional Bonded Particle Model
2004 Potyondy and Cundall Clustered Particle Model
2007 Cho et al. Clumped Particle Model
2010 Potyondy Grain-Based Model
2012 Potyondy Flat Joint Contact Model
3.3.2.1 The Conventional Bonded Particle Model (CBPM)
Potyondy and Cundall (2004) developed conventional bonded particle model, in which circular
or spherical particles bonded at their contact points using contact or parallel bonds to behave like
a brittle material such as rock. A contact bond applies at the contact point, while a parallel bond
acts over a finite area between two particles. Figure 3.3 shows contact and parallel bond models
in PFC. Developing a BPM using parallel bonds cannot generally match large ratios of UCS to
tensile strength. In this model, the interface is bonded across its entire length and when the bond
between particles breaks, the parallel bond is removed and the interface no longer resists relative
rotation. Cho et al. (2007) used parallel bond model to simulate Lac du Bonnet granite and
demonstrated that the highest achievable compression to tensile strength ratio in a parallel
bonded material is 4.
32
Diederichs (2000) developed BPMs using contact bonds between particles. He simulated several
rock strength tests using the BPMs and identified the following limitations: low compressive to
tensile strength ratio, linear failure envelope, underestimation of the confined strength (low
friction angle), and the ductile post peak response.
Figure 3.3: Illustrations of contact and parallel bond models in PFC (after, Potyondy and
Cundall, 2004).
3.3.2.2 The Clustered Particle Model (ClsPM)
To address the limitations of BPM mentioned in the previous section and better simulate the
irregular shapes of rock grains, Potyondy and Cundall (2004) introduced clustered particle model
by attaching a number of particles with finite or infinite bond strength in order to model
breakable or unbreakable grains (Figure 3.4). They showed that the friction angle can be
increased and the target confined strength can be matched using the ClsPM. Cho et al. (2007)
used CLsPM for simulation of Lac du Bonnet granite and the results showed that the minimum
ratio of UCS/tensile strength increases to 8 using the ClsPM, however, this ratio is still low
compared to the one for Lac du Bonnet granite that is more than 20.
3.3.2.3 The Clump Particle Model (ClmPM)
Cho et al. (2007) introduced an approach called clump particle model in which the clusters are
rigid (Figure 3.4). The irregular shape of a rock grain can be captured more realistically with
33
clumps. They showed that the ratio of UCS/tensile strength remains close to 14 and there is an
excellent agreement between the failure envelope predicted by the ClmPM and that of the LdB
granite. The limitation of this approach is the assumption that the grains are unbreakable which
may not be realistic in a compression test.
Figure 3.4: Particle rotation mechanisms in clustered (right) and clumped particles (left) (after,
Choe et al., 2007).
3.3.2.4 The Grain-Based Model (GBM)
Potyondy (2010) developed a GBM in which deformable, breakable or unbreakable grains
bonded according to the smooth joint contact model (Figure 3.5). Using the GBM with the
unbreakable grains, he was able to match the laboratory response of Ӓspӧ diorite rock subjected
to direct tension, unconfined and confined compression tests. Bahrani et al. (2012) also showed
that using a GBM, the previous limitations of the BPM were resolved in the calibration of
Wombeyan marble. However, result of the 2D GBM with unbreakable grains overestimated the
Brazilian tensile strength of compact rock, (Bahrani et al., 2012). Additional studies are needed
to explore the effects of grain breakage on the GBM behavior. Most of these models (i.e. ClsPM,
ClmPM, and GBM) have been developed in two-dimensional BPM and yet to be extended to
three-dimensional modeling.
34
Figure 3.5: GBM consisting of grains (bonded disks) and interfaces (smooth-joint contacts)
(after, Potyondy, 2010).
3.3.2.5 The Flat Joint Contact Model
Potyondy (2012a) developed a flat joint contact model so that each contact simulates the
behavior of a finite-length interface between two particles with locally flat notional surfaces
(Figure 3.6). The interface is segmented and each segment initially bonded. As the bonded
segments break, the interface behavior evolves from a fully bonded state to a fully debonded and
frictional state. Since the flat joint is not removed, even a fully broken interface continues to
resist relative rotation. The continued moment-resisting ability is an important microstructural
feature of this model that makes it possible to match the relatively high ratio of UCS to tensile
strength of rock, (Potyondy, 2012b).
35
Figure 3.6: 2D flat-joint contact (left) and flat-jointed material with effective surface of one grain
highlighted (right) (after, Potyondy, 2012a).
3.3.3 Model Generation
The intact rock was simulated as a bonded particle model (BPM) using Itasca’s 2D Particle Flow
Code (PFC2D). The material genesis procedure starts with the creation of a material vessel with
its dimensions conforming the dimensions of laboratory test specimens (12 cm height and 5 cm
of width). The particles in the model were generated randomly in the material vessel to match the
target porosity of the Meta-sandstone intact rock (5%) with a minimum particle size of 0.3 mm.
The density and local damping coefficient were set to the particles to efficiently remove kinetic
energy from the system. The system is then subsequently solved to satisfy an equilibrium
criterion and achieve a dense packing of circular particles. This is followed by bonding all
contacts between circular particles using the recently developed flat-joint contact model,
(Potyondy, 2012). Figure 3.7 shows the BPM generated to simulate the intact rock samples using
a flat-joint contact model. The same figure shows a close-up view of the rigid circular balls
bonded with flat-joint contacts.
36
Figure 3.7: Simulation of an intact rock specimen using BPM in PFC2D and effective interface
geometry of flat-joint contact model between balls.
3.3.4 Model calibration with static and dynamic properties of intact rock
The laboratory test results provided by CanmetMINING laboratory were used to calibrate the
bonded particle model that simulates both static and dynamic properties of the intact rock. A
bonded particle model (BPM) is characterized by its particle density, particle size distribution, as
well as by the assembly and micro-properties of particles and contacts used in the model. The
inverse calibration method which utilizes a trial and error process, was used to establish the
necessary micro-properties which consist of strength and stiffness parameters of particles and
contacts that simulate the laboratory intact rock material. To minimize the number of iterations, a
systematic calibration procedure was undertaken. The procedure started with the calibration of
elastic constants. For the simulation of a UCS test, the bond strength is set to a large value to
prevent bond failure and thereby to force the material to behave elastically. The Young’s
modulus is matched by varying the elastic modulus of the flat-joint contacts. Then the Poisson’s
ratio is matched by varying the ratio of the contact normal to shear stiffness. A few iterations are
needed to match both values. Once the desired elastic response is obtained, the flat-joint contact
tensile strength and cohesion parameters are changed to match the UCS and Brazilian indirect
tensile strengths. During this calibration process, the contact friction angle is set as zero. The
bond friction angle could be chosen to match the confined compressive strength and the particle
37
friction coefficient can be varied to reproduce post-peak behavior, but such parameters were
outside the scope of this study.
In a PFC2D model, the particle size cannot be regarded as a free parameter that only controls
model resolution; the elastic constants (e.g. the Young’s modulus and Poisson's ratio) and the
unconfined compressive strength are almost independent of particle size. However, the Brazilian
strength exhibits a clear dependence on particle size. The Brazilian tensile strength decreases as
the particle size is reduced. In this sense, the particle size cannot be chosen arbitrarily, (Potyondy
and Cundall, 2004). For the calibration process, a minimum particle size of 0.3 mm was chosen,
which is close to the minimum grain size of the rock test specimens (0.1 mm).
Multiple combinations of micro-properties can create a well-calibrated model which is not a
unique model. The lack of uniqueness is not necessarily a limitation because combinations of
different material parameters in nature can lead to similar behaviors in terms of failure
mechanisms and strength. Moreover, determination of micro-properties of rock is very difficult if
not impossible.
The UCS test is simulated by moving the top and bottom boundaries in order to load the BPM.
During the test, the vertical forces acting on the boundaries and their displacement are recorded.
This allows the computation of the axial stress and strain. Figure 3.8.a shows the stress-strain
curve for the resulting calibrated intact rock specimen. Elastic moduli (Young’s modulus and
Poisson’s ratio) are calculated for the linear portion of the stress-strain curve at 50% of ultimate
specimen strength. Figure 3.8.b shows the initiation and propagation of micro cracks during the
compression test. The micro-cracks are shown in red color.
38
(a (b)
Figure 3.8: (a) Stress-Strain response from a compression test; (b) Micro-crack distribution
within the specimen.
To simulate a Brazilian test, the specimen is trimmed into a disk shape and set in contact with the
loading platens. Then, the specimen is loaded by moving the boundaries toward one another.
During the test, the average force acting on the boundaries is monitored, and the maximum value
is recorded to calculate the indirect tensile strength of the rock. Figure 3.9 shows the micro-
cracks developed within the BPM specimen at the end of the Brazilian test. Micro-cracks are
shown in red color. In this case, tensile splitting is the macroscopic failure mode, as shown in
Figure 3.9 which includes a major tensile fracture parallel to the loading direction at the center of
the specimen as well as some fracture branching close to the diametral plane.
Figure 3.9: Micro-crack distribution within a BPM during a simulated static Brazilian test.
39
To simulate a dynamic drop test with PFC2D, the specimen is trimmed into a disk shape and put
in contact with a frictionless rigid platen at the bottom. The upper drop plate is simulated using
the clump logic. The properties of the drop plate clump are summarized in Table 3.3. Figure 3.10
shows the simulation of a drop test with PFC2D. The specimen is loaded by dropping the plate
toward the specimen. During the test, the impact force acting on the specimen is monitored and
the maximum impact force is recorded to calculate the dynamic tensile strength of rock.
Figure 3.10: Drop test simulation with PFC2D (left) versus laboratory test (right).
Table 3.3: Clump properties for simulating the drop plate
Local damping factor 0.1
Weight 617 kg/m
Impact velocity 3 m/s
Normal to shear stiffness ratio 1
Friction coefficient 0.5
For calibration purposes, the properties of the contact between the drop plate and the specimen
are established so that it results in the same impact force on the specimen that was recorded in
the laboratory. Itasca (2014) suggested that in a dynamic impact simulation with PFC, it may be
necessary to set the local damping coefficient to a low value and implement viscous damping to
obtain realistic energy dissipation. Therefore, low local damping of 0.1 was assigned to the
40
particles in the specimen and to the clump. A linear contact model is assigned between the
hitting clump and the model particles to satisfy the viscous damping conditions. The linear
contact model provides linear spring and dashpot components that act in parallel with each other.
The linear springs provide linear elastic frictional behavior with constant normal and shear
stiffness, while the dashpot component provides viscous behavior with the normal and shear
critical-damping ratios. Figure 3.11 illustrates the components of a linear contact model.
Figure 3.11: Components of linear contact model (after, Itasca, 2013).
Table 3.4 summarizes the properties assigned to the contact between the drop plate and the
simulated specimen in order to calibrate the test results. The result of the initiation and
propagation of micro-cracks due to the drop test is shown in Figure 3.12.
Table 3.4: Micro-parameters assigned to the contact between the drop plate and the specimen
Normal damping ratio (βn) 1
Shear damping ratio (βs) 1
Elastic Modulus (GPa) 60
Normal to shear stiffness ratio (Kn/Ks) 1
41
Figure 3.12: Micro-cracks developed within the BPM after the drop test.
The test enabled the determination of the dynamic tensile strength. The determined dynamic
tensile strength is higher than the static ones because of the complexity of the dynamic fracture
process (as explained in detail in section 2.3). Comparison between Figure 3.9 with 3.12 shows
the rock sample fails with more fractures and broken parts under the dynamic loading.
Table 3.5 summarizes all the micro-properties of the BPM obtained through the calibration
process. The calibrated model shows a good agreement between the experimental and numerical
results with error levels less than 10% as shown in Table 3.6. The maximum error percentage
was recorded for Poisson’s ratio while the Young’s modulus has the minimum error percentage.
Table 3.5: Micromechanical properties of the calibrated bonded particle model (BPM)
Ball Properties
Density (kg/m3) 2713
Minimum radius (mm) 0.3
Maximum to minimum radius ratio 1.66
Local Damping Factor 0.1
Contact Properties
Contact gap (m) 0.5e-4
Elastic modulus (GPa) 65
Normal to shear stiffness ratio 3.3
Friction coefficient 0.5
42
Friction angle (º) 0
Tensile strength (MPa)
Mean 35
S.D 25
Cohesion (MPa)
Mean 120
S.D 50
Table 3.6: Comparison between experimental test results and the BPM simulation
Type of Test Mechanical Property Experimental PFC2D Error
%
Static Unconfined compression
test
Young’s Modulus (GPa) 74.5 74.9 0.5
Poisson’s ratio 0.26 0.24 7.7
Axial strain at peak stress
(%) 0.397 0.404 1.8
UCS (MPa) 267 270 1.1
Static Brazilian test
Static tensile strength (MPa) 21.4 22.3 4.2
UCS/Tensile strength 12.5 12.1 3.2
Drop test Dynamic tensile strength
(MPa) 29.82 30.3 1.7
43
3.4 Summary
This chapter presented the simulation and calibration of static and dynamic mechanical
properties of an intact rock. Meta-sandstone rock samples from a quarry in Nova Scotia were
collected and subjected to a series of laboratory tests including static uniaxial compression and
static and dynamic Brazilian tests at CanmetMINING Rock Mechanics Laboratory. The
mechanical properties of the intact rock material including Uniaxial Compressive and indirect
Tensile Strength, Elastic Modulus and Poisson’s Ratio, and the dynamic indirect Tensile
Strength of rock samples were provided by Labrie (2013). The distinct-element particle flow
code (PFC2D) was successfully used to generate a bonded particle model and simulate both the
static and dynamic mechanical properties of the intact rock. The bonded particle model was then
calibrated against experimental laboratory results. Using a flat-joint contact model enabled
matching a relatively high ratio of UCS to tensile strength for the intact rocks (12.5 experimental
vs. 12.1 numerical) which was a major limitation of the PFC code over the past decade.
In next chapter, the calibrated model was used to develop jointed rock mass samples by
incorporating discrete joint networks into a large-scale bonded particle model. This was followed
by conducting impact tests on the large-scale jointed rock mass samples for quantification of the
damage inflicted by rigid projectile particle.
44
Chapter 4 Impact-induced Damage of Jointed Rock Masses
4.1 Introduction
This chapter aims to study the impact-induced damage of jointed rock masses using a numerical
approach. The collision of a rock fragment with a rock initially leads to the closure of pre-
existing joints. This is followed by elastic deformation of the rock mass in the area of impact,
creating a crater zone, crushing the rock material, and creating cracks in the material. The impact
loading causes shear and tension cracks to be generated and propagated inside the rock mass. In
this study, damage was defined as the initiation and growth of micro-cracks within the numerical
models.
This chapter presents the results of numerical impact tests using the distinct element method
(DEM), in particular the Particle Flow Code (PFC). The calibrated intact rock model was used to
develop large-scale jointed rock mass sample units by incorporating discrete fracture networks
into the bonded particle model. Then, a rigid particle was projected against the jointed rock mass
samples and the extent of damage zone inflicted by the projectile particle on the rock mass
blocks was quantified. The extent of the damage inflicted by impact loading of the jointed rock
mass is influenced by the rock mass characteristics, impact condition, and type of projectile
particle. A parametric analysis was conducted to investigate the influence of the fracture
intensity and orientation within the rock mass, the impact velocity, impact angle, and projectile
particle density on the impact-induced damage of jointed rock masses.
45
4.2 Simulation of Jointed Rock Mass
In order to simulate a jointed rock mass, the intact rock model is calibrated for both static and
dynamic properties. It is then used to develop large-scale jointed rock mass samples by
incorporating discrete fracture networks into the bonded particle model. This allows for the
generation of 2D synthetic rock mass (SRM) models in which the bonded particle model
represents the rock material properties and a smooth-joint contact model (SJM) represents the
joint network, (Mas Ivars et al., 2011). The properties of the synthetic rock mass are controlled
by the combined behavior of the solid rock matrix, represented by a bonded particle model, and
the integrated fracture fabric, represented by a smooth-joint model.
The concept of the smooth-joint contact model was first proposed by Mas Ivars et al., (2008).
The smooth-joint contact model simulates the behavior of a joint surface, regardless of the local
particle contact orientations along the interface (Figure 4.1). The smooth joint model is assigned
to the particle contacts along the joints. This allows associated particles to slide through each
other along the joint plane rather than being forced to move around one another.
Figure 4.1: Smooth joint geometry (after, Mas Ivars et al., 2008)
In this work, a large-scale block of jointed rock mass was considered. The dimension of the rock
mass block was selected such that the reflected wave from the boundaries would be weak enough
to induce damage, which means that the edge effect of the target can be neglected. The calibrated
bonded particle model, presented in chapter 3, was used to develop a large-scale rock mass
46
sample unit of 2 m width and 1 m height. The model was composed of 3,878,268 circular
particles with a minimum particle diameter of 0.3 mm. The ratio of maximum versus minimum
particle size was fixed at 1.66 in order to generate a uniform particle size distribution in the
model. The micro-mechanical properties listed in Table 3.5 were assigned to the particles and
bonds in the model.
In order to generate SRM samples, discrete fracture networks were simulated using smooth joint
contact model. The mechanical properties listed in Table 4.1 were used to characterize the joints
surface properties. The joints were assumed to be cohesionless and have zero tensile strength.
Table 4.1: Mechanical properties of smooth-joints within the SRM samples
Normal stiffness (N/m) 1011
Shear stiffness (N/m) 1010
Tensile strength (MPa) 0
Cohesion (N/m2) 0
Friction coefficient 0.58
4.3 Simulation of Impact Test on Jointed Rock Mass Samples
The SRM samples were used to simulate the impact tests on jointed rock mass samples. Large-
scale rock mass samples were subjected to impact loads in order to investigate the effect of rock
mass joint characteristics, impact condition, and projectile particle property on the impact-
induced damage of the jointed rock mass. Thus, a parametric analysis was carried out to assess
the effect of joint intensity and joint orientation, impact velocity, impact angle, and projectile
particle density.
47
In the undertaken numerical models the rock fragment was assumed as a rigid particle; however
in reality, the projectile rock fragments are deformable and may break when they hit the target.
In PFC2D, the basic particle shape is modeled as a circular entity, but it is possible to simulate
different shapes by grouping circular particles together and using cluster or clump model. For the
purposes of this study, circular rock fragment shapes were used. Density and velocity values
were assigned to the generated rock fragment and it was projected against the SRM samples.
During the simulations, the following parameters were monitored: velocity of particle, impact
force, impact-induced stress, and number of micro cracks.
4.3.1 Impact Force on the Rock Mass Samples
The impact force on the jointed rock mass target due to a single particle impact is directly
proportional to the particle velocity and particle weight, and is inversely proportional to impact
duration, (Hambley et al., 1983). The impact forces on the SRM samples were measured in all
simulations. The impact force on the target depends on the contact properties between the
projectile particle and the target. Contact properties between the projectile particle and SRM
sample was assigned as listed in Table 3.4 and these values were based on calibration against
laboratory data for dynamic tensile strength of intact rock.
4.3.2 Impact-induced Stresses on the Rock Mass Samples
The impact of a projectile particle hitting the rock mass induces stress on the target. In the PFC
models, stress can be measured by averaging the contact forces between particles in the area
specified by a measurement circle. To measure the magnitude of stresses around the impact
point, six measuring circles were placed with the center at the impact point on the target and
with different radius ranging from 0.1 m to 1 m. Figure 4.2 presents the location of the
measuring circles within the rock mass samples. The magnitude of vertical stresses inflicted on
the target by the collision of the projectile particle was calculated within the measuring circle
zones in each impact test.
48
Figure 4.2: Measuring circles of different radii within the rock mass samples to compute the
impact-induced stresses.
4.3.3 Impact-induced Damage of the Rock Mass Samples
The collision of a rock fragment with the jointed rock mass samples creates a damage zone
within the rock masses. In this study, the initiation and growth of the impact-induced micro-
cracks were considered as quantitative parameters for rock mass damage assessment, as
proposed by Kachanov (1986). The shear and tensile cracks are generated inside of the rock mass
as a result of impact-induced damage.
4.3.4 Energy Transformations during the Impact
Energy transformation occurs during “approach” and “restitution” stages during the impact. The
approach stage is defined as the period from the beginning of impact to the maximum
penetration of the projectile particle within the target. During the approach period, the system's
kinetic energy decreases while the projectile particle penetrates into the target and it reaches
approximately zero when the particle attains its largest penetration, at which point almost all of
the system kinetic energy is transformed into strain energy. The restitution stage is the period
between the maximum penetration and the time of loss of contact between the projectile particle
and the target. During the restitution period, the stored strain energy in the springs is again
gradually transformed into system kinetic energy, which is less than that of the initial one, as part
of the energy has been dissipated by the dashpots and frictional slip.
49
There is a balance between the magnitude of kinetic energy that the projectile particles lose
during impact and the energy that is dissipated. The majority of the dissipated energy is absorbed
by the target rock while a small part of it is converted into other forms of energy such as
fragmentation of the projectile particle, heat, and sound. Energy loss is one of the most important
mechanisms in the understanding of the impact-induced damage mechanism.
The loss of kinetic energy causes the rebound velocity of the projectile particle to be less than the
impact velocity. The ratio between the magnitudes of the rebounding and impacting velocities of
the particle is defined as “coefficient of restitution”. The coefficient of restitution of rock
fragments falling on a surface depends on a variety of factors, including the size, shape, and type
of the rock fragments, the geometry of the surface, the velocity of the rock fragments and the
impact angle, (Azzoni and deFreitas, 1995). The rock drop test is the most popular test that can
be done in the laboratory (Chau et al., 2002; Imre et al., 2008) and in the field (Azzoni and de
Freitas, 1995) to measure the coefficient of restitution. However, the laboratory tests do not take
into account the rock fractures while in-situ tests are very expensive.
4.4 Effect of Joint Characteristics on Impact-Induced Damage of
Rock Masses
The key characteristics that could control the impact-induced damage of a rock mass are the
intact rock properties and its joint characteristics. Experience shows that presence of
discontinuities in the rock mass can considerably increase the extent of impact-induced damage.
In addition, the geometrical characteristics of discontinuities have a great influence on the
erosion rate of rock mass, (Hadjigeorgiou et al., 2005). Despite the experimental and
observational insights of this issue, limited studies have been conducted to quantify the impact-
induced damage of jointed rock masses.
Most recently, Esmaieli and Hadjigeorgiou (2014) conducted a series of numerical experiments
to investigate the influence of rock mass foliation on impact-induced damage of ore pass walls.
The PFC2D code was used to simulate three distinct rock masses (5m x 5m) surrounding an ore
pass, characterized by different foliation angles. Subsequently, a rock fragment was projected
against the ore pass walls at a constant impact angle and velocity. The investigation
demonstrated that the presence of foliation planes had a significant influence on the size of the
50
damage zone and results in wider and deeper damage in a jointed rock mass. This implies that
the foliations act as weakness planes which accelerated the impact-induced crack formation. It
was also shown that the foliation orientation has an influence on the extent of impact-induced
damage to the ore pass walls. Rock mass with foliation planes oriented semi-parallel to the ore
pass wall result in greater damage than the ones with foliation planes oriented perpendicular to
the walls. This investigation highlighted the importance of driving the ore passes against the
foliation dip.
To assess the effect of joint characteristics on the impact-induced damage of jointed rock
masses, the following sections present the result of numerical impact tests conducted on the rock
mass samples with different joint intensities and orientations.
4.4.1 The Effect of Joint Intensity
To study the effect of joint intensity on the impact-induced damage of rock masses, rock mass
blocks with different joint intensities were generated. The cumulated length of joint per total area
of rock block (P21) was considered to be representative of the rock mass joint intensity in the 2D
simulations. In order to generate SRM samples with different joint intensities (P21), three discrete
joint networks with joint intensity (P21) of 1, 2, and 3 m-1 and with the same mechanical
properties as those listed in Table 4.1 were embedded into the large-scale bonded particle model
of 2 m x 1 m. The resulting generated SRM samples with different joint intensity are shown in
Figure 4.3. A 0.4 m diameter rock fragment with the density of 2700 kg/m3 was generated above
the rock mass samples and was projected vertically against the SRM samples with an impact
velocity of 15 m/s (Figure 4.4).
51
Figure 4.3: Jointed rock mass samples with different joint intensities.
Figure 4.4: Impact test on the rock mass samples.
52
Figure 4.5 presents the damage inflicted by the rock fragment projected on the jointed rock mass
samples. For comparison purpose, the damage inflicted to the large-scale intact rock sample with
no joint was also presented in Figure 4.5 a.
(a)
(b)
(c)
53
(d)
Figure 4.5: Impact-induced damage within the rock mass samples with different joint intensities:
a) P21 = 0 m-1; b) P21=1 m-1; c) P21 = 2 m-1; d) P21 = 3 m-1. Tension and shear cracks are indicated
in red and black, respectively.
The results of numerical impact tests carried out on the SRM samples demonstrate that shear and
tension impact-induced micro-cracks were generated within the rock mass. The majority of the
impact-induced micro-cracks were generated in tension. For all the rock mass models analysed,
the crushed zone can be seen in the vicinity of the impact point. The size of the crushed zone is
almost the same for all rock mass samples. Around the crushed zone, a high-density micro-crack
zone is generated while at a distance from this region, discrete micro cracks are propagated.
Finally, at a distance from this region, median and radial cracks are initiated and propagated.
Figure 4.5 clearly shows the dependency of the micro-crack patterns on rock mass joint intensity.
As the rock mass joint intensity increases, the crack pattern and consequently the impact-induced
damage inflicted to the rock mass becomes wider and deeper. In addition, as the joint intensity
increases, the radial and side cracks propagate longer and deeper within rock mass sample. For
the rock mass sample with a low joint intensity, P21 = 1 m-1, crack propagation stops when it
reaches the pre-existing joint below the surface. For the two SRM samples with higher joint
intensity, P21 = 2 and 3 m-1, other cracks radiate from the initial radial cracks or from pre-existing
joint surfaces. These cracks propagate almost perpendicular to the pre-existing joint surface.
Figure 4.6 shows the relationship between the number of impact-induced micro-cracks and the
rock mass joint intensity. As the joint intensity (P21) of SRM samples increases up to a certain
54
level, more micro cracks are initiated and propagated due to impact. However, for the SRM
samples with P21 of 2 and 3 m-1, almost the same numbers of micro-cracks have developed.
Figure 4.6: Influence of rock mass joint intensity on the number of impact-induced micro-
cracks.
The normal impact force acting on the SRM samples is 28 MN. The magnitudes of average
vertical stresses at different distances from the impact point are shown in Figure 4.7. The
magnitude of stresses at the impact region is high and decreases with an increase of distance
from the collision point. Results show that the effect of impact-induced stresses becomes
negligible at 1 m from the impact point. This further supports the reason for the selected
dimension of the rock mass block.
55
Figure 4.7: Impact induced stresses at different distances from the impact point within the rock
mass samples with different joint intensity.
Figure 4.8 shows the influence of rock mass joint intensity on the rebound velocity of the
projectile particle, and consequently the coefficient of restitution. By increasing the fracture
intensity, more energy is dissipated in the rock mass causing the development more fractures and
consequently, the rebound velocity decreases.
56
Figure 4.8: The effect of joint intensity on the coefficient of restitution.
4.4.2 The Effect of Joint Orientation
The impact-induced damage of SRM samples depends not only on the intensity of joints in the
rock mass but also on their orientation. For this reason, three SRM samples with joint intensity
of P21 = 3 m-1 and different orientation including vertical, horizontal, and 45º dip angle were
generated. The same rock fragment with a diameter of 0.4 m, density of 2700 kg/m3, and
velocity of 15 m/s, was projected vertically against the SRM samples. The results of numerical
impact tests carried out on the SRM samples with different joint orientations are shown in Figure
4.9.
57
Figure 4.9: Jointed rock mass samples with different joint orientations.
Figure 4.10 clearly shows the dependency of the micro-crack patterns on the orientation of pre-
existing joints. The damage pattern forms as a layer of heavily crushed area around the impact
point from which several radial (wing) cracks extend. The radial cracks generally stop when they
reach the pre-existing joints and are almost perpendicular to the joint surface. This investigation
emphasises that excavations under impact loading must be driven against joint orientation. The
impact point with respect to the pre-existing joint location on the surface of rock mass sample is
also of particular importance in the extent of damage and micro-crack pattern.
58
(a)
(b)
(c)
Figure 4.10: Impact-induced damage within rock mass samples with different joint orientation:
a) dip = 0º; b) dip =45º; c) dip = 90º; Tension and shear cracks are indicated in red and black,
respectively.
59
Figure 4.11 shows the relationship between the number of impact-induced micro-cracks and rock
mass joint orientation. The number of impact-induced micro cracks on the rock mass block with
horizontal joints is significantly greater than that of the vertical and inclined ones. The pre-
existing joints restrain the micro-cracks from being propagated in the rock samples with vertical
and inclined joints; however, the micro-cracks have more space to propagate laterally in rock
sample with horizontal joints.
Figure 4.11: Influence of joint orientation on the number of impact-induced micro-cracks.
Figure 4.12 shows the magnitude of vertical stresses in rock mass samples at different distances
from the impact point. As shown in the figure, the change in the stress field due to the joint
orientation is negligible.
60
Figure 4.12: Impact induced stresses at different distances from the impact point within the rock
samples with different joint orientations.
The effect of joint orientation on the coefficient of restitution is presented in Figure 4.13. The
sample with horizontal fractures has higher energy dissipation, which caused greater number of
impact-induced micro-cracks and consequently less rebound velocity.
Figure 4.13: The effect of joint orientation on the coefficient of restitution.
61
4.5 Effect of Impact Condition on Impact-Induced Damage of
Rock Masses
The damage inflicted on a rock mass depends on the impact condition including impact velocity
and angle. Researchers have investigated the effects of impact velocity via experimental tests.
Hutchings (1992) reported that the extent of impact damage depends on the number and mass of
individual boulders striking the surface as well as the impact velocity. Goodwill et al. (1999)
suggested that erosion wear in ore passes is roughly proportional to the impact velocity raised to
the power of 2.5. As discussed in section 2.4.1, Xia and Ahrens (2001) carried out laboratory
impact tests on San Marcos gabbro targets using aluminum bullets having a velocity range of
800-1200 m/s and measured the depth of damage. They showed that increasing the impact
velocity causes greater damage depth. Momber (2003) also carried out particle impact
experiment as discussed in section 2.4.1. He also measured the damage rate of the rock targets
based on different particle velocities. He found that the velocity of particles hitting a rock sample
accelerates the extent of inflicted damage.
Natural impact events generally happen at impact angles less than vertical. However, for
simplification of the problem a large amount of theoretical and experimental works, in the
impact cratering field, have been performed under normal impact conditions. Gault and
Wedekind (1978) conducted a comprehensive study and indicated that vertical hypervelocity
impacts are regarded as a good representation of oblique impacts since oblique hypervelocity
impacts with impact angles higher than 30º produce circular craters similar to that observed in
vertical hypervelocity impacts. Cao et al. (2011) conduct a series of impact experiments on
granite rock plate by low velocity ball bullet using Split Hopkinson pressure bar. The impact
tests were carried out with different bullet speeds in the range of 5-35 m/s and impact angles of
5º to 45º. According to their experimental results, increasing the impact velocity creates greater
break area and crack length. They also showed that by increasing the impact angle, the crushed
zone gradually increases.
In the following sections, a series of numerical tests were conducted to better understand the
effect of impact velocity and impact angle on the extent of impact-induced damage on the rock
mass samples.
62
4.5.1 The Effect of Impact Velocity
Using the generated jointed rock mass sample with the joint intensity of P21 = 3 m-1, numerical
impact tests were conducted to investigate the influence of impact velocity on the damage
inflicted by the particle impact. The same rock fragment with diameter of 0.4 m and density of
2700 kg/m3 was projected vertically against the SRM samples with impact velocity of 5, 10, and
15 m/s (Figure 4.14). The impact velocities were selected based on previous investigations of
material flow simulations for ore pass systems (Esmaeili and Hadjigeorgiou, 2011). The results
of impact tests on the SRM samples are shown in Figure 4.15.
Figure 4.14: Impact test on the jointed rock mass sample.
63
(a)
(b)
(c)
Figure 4.15: Impact-induced damage conflicted on the rock mass sample by different impact
velocities: a) V = 5 m/s; b) V =10 m/s; c) V = 15 m/s; Tension and shear cracks are indicated in
red and black, respectively.
64
Figure 4.15 shows that increasing the impact velocity results in a wider and deeper damage zone
in the rock mass samples. The effect of projectile particle velocity on the number of impact-
induced micro-cracks in the rock mass blocks is presented in Figure 4.16. As the impact velocity
increases, more micro-cracks propagate in the rock mass. This is due to the greater impact force
and vertical stresses recorded at higher impact velocities which are presented in Figure 4.17 and
4.18, respectively.
Figure 4.16: Influence of impact velocity on the number of impact-induced micro cracks.
Figure 4.17: The effect of impact velocity on the acting force on the rock mass samples.
65
Figure 4.18: Impact-induced stresses at different distances from the impact point within the rock
mass samples subjected to different impact velocities.
Figure 4.19 shows the dependency of the coefficient of restitution on the impact velocity. As
shown in the figure, at higher impact velocities, more energy is dissipated in the rock mass
sample causing greater damage in terms of the number of impact-induced micro cracks. As a
result, the rebound velocity of the rock fragment decreases with the increase of impact velocity.
Experimental studies performed by Ushiro et al. (2000) confirm the same relationship between
the coefficients of restitution and impact velocity.
Figure 4.19: The effect of impact velocity on the coefficient of restitution.
66
4.5.2 The Effect of Impact Angle
Using the generated SRM sample with the joint intensity of P21 = 3 m-1, a rock fragment with
diameter of 0.4 m, density of 2700 kg/m3, and velocity of 15 m/s was projected against the
sample with impact angles of 30º, 60º, and 90º as shown in Figure 4.20. The impact angle is
measured from a plane tangent to the impact surface. Figure 4.21 shows the impact test results on
the SRM sample.
Figure 4.20: Impact test on the jointed rock mass sample with different impact angles.
67
(a)
(b)
(c)
Figure 4.21: Impact-induced damage conflicted on the rock mass sample by different impact
angles: a) α = 30º; b) α =60º; c) α = 90º; Tension and shear cracks are indicated in red and black,
respectively.
68
Figure 4.21 shows that by decreasing the impact angle, the mechanism of collision between the
projectile particle and its intended target changes from impact to frictional sliding. The impact
test with angle of 30º causes a small local crack zone in the rock mass block. By increasing the
impact angle, more radial and median cracks propagate through the sample. Unlike the crack
patterns observed with the impact tests conducted at an angle of 30° and 60°, near symmetric
micro-crack patterns were observed under vertical impact tests. Most of the micro-cracks still
developed perpendicular to the pre-existing joints regardless of the impact angle.
Figure 4.22 illustrates the relationship between the impact angle of the projectile particle and the
number of micro-cracks that are produced in the rock mass block. As the impact angle increases,
more micro-cracks are induced in the rock mass due to the greater vertical force and stress.
Figure 4.22: Influence of impact angle on the number of impact-induced micro-cracks.
The normal and shear impact forces acting on the rock mass sample for different impact angles
are presented in Figure 4.23. By increasing the impact angle, higher normal force and less shear
force are applied on the target. Figure 4.24 presents the magnitude of the vertical stresses at
different distances from the impact point within the rock mass sample. Increasing the impact
angle causes higher vertical stresses to be recorded in the vicinity of the impact point.
69
Figure 4.23: The normal and shear impact force acting on the rock mass sample for different
impact angles.
70
Figure 4.24: Impact induced stresses at different distances from the impact point within the rock
sample subjected to different impact angles.
Figure 4.25 presents the normal and tangential coefficient of restitutions for the impact tests of
jointed rock mass subjected to different impact angles. The results show that as the impact angle
on the rock mass surface increases, the rebound velocity of the projectile rock bolder decreases.
Experimental studies carried out by Wu, (1985), Wong et al., (2000), and Chau et al., (2002) also
indicated the same correlation between the coefficients of restitution and the impact angle. Those
studies point out that increasing the impact angle could decrease the coefficient of restitution.
Figure 4.25: The effect of impact angle on the coefficient of restitution.
71
4.6 Effect of Projectile Particle Properties on Impact-Induced
Damage of Rock Masses
The extent of impact damage depends on the size and mass of rock fragments striking the rock
mass surface. The results of numerical experiments by Hadjigeorgiou and Esmaieli (2012)
indicate that there is a correlation between the extent of impact induced damage on the ore pass
walls and the size of projectile particles. They showed that larger rock fragments could cause
more damage on the ore pass walls. In the following sections, the effects of projectile particle
density on the impact-induced damage of rock mass were investigated.
4.6.1 The Effect of Projectile Particle Density
To investigate the effect of projectile particle density, rock fragments of 0.4 m of diameter with
three different densities of 2100, 2700, and 3000 kg/m3 were projected vertically against the
SRM sample of P21= 3 m-1 with an impact velocity of 15 m/s (Figure 4.26). Figure 4.27 presents
the damage inflicted by rock fragments with different densities projected on the jointed rock
mass samples.
Figure 4.26: Impact test on the jointed rock mass sample with projectile particle of different
densities.
72
(a)
(b)
(c)
Figure 4.27: Impact-induced damage inflicted on a rock mass sample with a projectile particle of
different densities: a) d = 2100 Kg/m3; b) d =2700 Kg/m3; c) d = 3000 Kg/m3; Tension and shear
cracks are indicated in red and black, respectively.
73
The number of impact-induced micro-cracks inflicted on the SRM sample is presented in Figure
4.28. As shown in Figure 4.29, rock fragments with a higher density apply a greater impact force
on the rock mass sample and consequently cause greater number of micro-cracks in the sample.
Figure 4.30 shows the average vertical stress at different distances from the impact point within
the rock mass sample. The results also indicate that at the vicinity of the impact point, higher
stresses are developed with higher projectile particle mass.
Figure 4.28: Influence of projectile particle density on the number of impact-induced micro
cracks.
Figure 4.29: The normal impact force acting on the rock mass sample for impact tests
with different projectile particle density.
74
Figure 4.30: Impact induced stresses at different distances from the impact point within the rock
mass sample with different projectile particle densities.
4.7 Summary
The PFC2D code was used to simulate the impact of a single projectile particle on jointed rock
mass samples. The rock mass itself was simulated using the synthetic rock mass approach by
embedding a discrete joint network into a large-scale bonded particle model. Damage inflicted
by a single projectile particle thrown against the rock mass samples was quantified based on the
number of micro-cracks initiated and propagated due to the impact. The impact-induced micro-
cracks were generated in the forms of tensile and shear failure. The damage pattern was modeled
as an area of heavily crushed rock around the impact point, from which several radial (wing)
cracks extend. For all simulations, impact force, impact-induced stress, number of impact
induced micro-cracks, and particle rebound velocity were recorded.
The extent of the damage inflicted by impact loading on the jointed rock mass samples is
influenced by the joint characteristics, impact conditions, and projectile particle properties. The
capacity of the rock mass to resist impact loads is determined by the joint characteristics. To
investigate this effect, rock mass samples with different joint intensities and orientations were
75
generated. The numerical impact test results show that the presence of structural defects within a
rock mass plays a very important role in the magnitude of impact-induced damage inflicted to
the rock mass. It is recognized that higher intensity of joints within the rock mass results in more
pronounced rock mass damage. Furthermore, the micro-crack patterns were also governed by the
joint orientation of the rock mass sample. It was also observed that radial cracks develop almost
perpendicular to the pre-existing joints.
Impact tests on the rock mass target were carried out using different impact angles and
velocities. The results indicate that higher impact velocities cause the greater extent of damage
due to the higher impact-induced stresses. The impact angle also has significant influence on the
inflicted damage, so that by increasing the impact angle, a wider and deeper damage zone was
created.
The effect of rock fragment density was also investigated. Projectile particles with higher density
applied greater impact force on the rock mass sample, and consequently, initiated more micro-
cracks in the SRM sample.
The numerical experiments described in this chapter have allowed to quantify the influence of
different factors including joint characteristics, impact condition and projectile particle
properties on the impact-induced damage of jointed rock masses.
76
Chapter 5 Conclusions and Future Work
5.1 Introduction
This final chapter provides a summary and conclusions of this research project, together with
recommendations for future research. An overview of the work undertaken in this thesis to better
understand the impact-induced damage of jointed rock masses is presented. This is followed by
the important conclusions drawn from the current work. Finally, perceived limitations of this
study are outlined and suggestions for the refinement and extension of the present work for
future research are provided.
5.2 Summary of the research work
The major contribution of this thesis was the development of a numerical approach to better
understand impact-induced damage of jointed rock masses. This main objective was divided into
secondary objectives including: 1) Simulate the static and dynamic properties of the intact rock,
2) Apply a numerical method that can quantitatively evaluate the impact-induced damage of
jointed rock masses, 3) Investigate the effect of different parameters such as joint characteristics,
impact condition, and projectile particle properties on the damage inflicted by impact on rock
mass.
To achieve these objectives, the following works were undertaken:
A review of previous work was undertaken and used to develop a strategy. The result of a series
of laboratory tests on intact rock specimens including both static and dynamic mechanical
properties were used to provide a reference for the undertaken work. The distinct element
particle flow code (PFC2D) was used to simulate both the static and dynamic mechanical
properties of the intact rock. Additionally, the bonded particle model was calibrated against
experimental laboratory results. Then, large-scale jointed rock mass samples (SRM samples)
were created by inserting joint networks into a bonded particle model of 2 m wide x 1 m high.
Finally, a rigid particle was projected against these jointed rock mass samples and the inflicted
77
damage was quantified based on the initiation and growth of micro-cracks within the SRM
models. A parametric analysis was performed to investigate the influence of joint intensity and
orientation, impact velocity, impact angle, and projectile particle density on the response of the
jointed rock mass to the impact load.
5.3 Conclusions
Numerical simulations were undertaken in order to evaluate the response of jointed rock masses
to impact loading. Based upon the results of these simulations, it was concluded that:
The bonded particle model was successfully used to simulate both the static and dynamic
mechanical properties of a Meta-sandstone intact rock. Using a flat-joint contact model
enabled matching relatively large ratio of UCS to tensile strength for the intact rock
which was a major limitation of the original PFC code over the past decade.
The 2D synthetic rock mass approach was adopted in this study. There was no need for a
user specified constitutive model assigned to intact rock and joints. The mechanical
behaviour of a SRM sample depends on the combined behavior of both the solid rock
matrix and the embedded joint network.
The results of impact tests on the SRM samples show that the presence of pre-existing
structural defects within a rock mass plays a very important role in the amount of impact-
induced damage inflicted to the rock mass. It was concluded that higher intensity of joints
within the rock mass results in more pronounced rock mass damage. The radial cracks
generally stop when they reach the pre-existing joints and they are almost perpendicular
to the joint surface. The impact-induced damage of SRM samples also depends on the
joint orientation. This investigation emphasises that the rock structures under impact
loading must be driven against joint orientation.
The extent of impact-induced damage on the rock mass samples depends on the impact
velocity and angle. The highest impact load, and consequently greatest damage were
recorded at higher impact velocity and impact angle near vertical of the projectile
particle.
78
The effect of projectile particle mass was investigated. Projectile particle with higher
density shows more initiation and growth of micro cracks in the rock mass.
Since field or laboratory impact tests on large-scale rock mass samples are expensive and
difficult to carry out, using numerical modelling provides a reasonably good
understanding of the impact-induced damage mechanism of large-scale rock mass
samples.
5.4 Limitation of the employed methodology
Some of the limitations of the employed methodology are summarized as follow:
The preference for 2D instead of 3D modeling in this work was motivated by shorter
execution times for solving models and overall simplicity. It is recognized that such
simplifications also have some limitations. The 2D rock mass models usually
underestimate rock mass mechanical properties. Additionally, the 2D model is not fully
representative of a jointed rock mass, it only presents a cross section of it.
In the present work, only circular particles were used as projectile rock fragments on the
jointed rock mass samples. In reality, the rock masses are subjected to the impact loading
of the fragments with various shapes. The particle shape affects the impact-induced stress
field in the rock mass and consequently the inflicted damage.
In the undertaken numerical models, the rock fragment was assumed as a rigid particle;
however, in reality the projectile rock fragments are deformable and may break upon
impact with hit the target.
5.5 Future work
The numerical approach employed in this study needs further refinement to precisely predict the
extent of damage zone in the jointed rock masses.
The 2D model was used as a preliminary analysis to evaluate the damage inflicted by
impact loading on the jointed rock masses due to the simpler geometry and shorter
79
execution time. The work could be extended to 3D to simulate this phenomenon more
realistically.
The analysis of rock mass damage, presented in chapter 4, assumes a single particle and it
does not consider the effect of multiple particle impacts or repetitive single particle
impact. The analysis can be further extended to consider the impact of a stream of
flowing particles on the rock mass.
The extent of the damage inflicted by impact loading to rock mass samples is also
influenced by shape of the projected rock fragment. The circular particle shape is
assumed in the numerical simulations. To consider shape effect, it is possible to simulate
different shapes by grouping circular particles together and using cluster or clump
models. This allows for the fragmentation of the projectile particle upon impact with the
target.
Parametric analysis can be extended to investigate the influence of other rock mass
parameters on the response of the rock mass to the rock fragments impact by analyzing
the effect of joint persistence, the effect of mechanical properties of joints and intact rock,
and the effect of stress states.
The work can be further extended to consider energy dispatching during rock mass
impact tests. All forms of energy can be tracked to see how the energy is transformed and
how this transformation is related to the failure process. The result of this investigation
can contribute to the better understanding of impact-induced rock mass fragmentation
which is commonly used in drills and crushers for rock breakage.
80
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Appendices
Appendix A: Laboratory Test Results
This appendix presents the laboratory test results provided by CanmetMINING laboratory for
metasandstone samples including static unconfined compressive test, static Brazilian test, and
drop test, which are summarized in chapter 3.
Table A.1: Laboratory static unconfined compression test results for meta-sandstone samples
Sample
No.
Density
(g/cm3)
Diameter
(mm)
Length
(mm)
Failure load
(kN)
UCS
(MPa)
Axial
Strain
(%)
Elastic
Modulus
(MPa)
Poisson's
Ratio
1 2.70 47.06 117.81 315.15 181.23 0.237 80041 0.278
2 2.67 47.13 116.93 546.48 313.17 0.415 80909 0.236
3 2.69 47.22 117.26 305.12 174.25 0.224 81201 0.219
4 2.71 47.16 116.23 540.48 309.37 0.479 71053 0.340
5 2.77 47.06 115.46 340.55 195.83 0.330 62989 0.269
6 2.68 47.06 116.03 382.44 219.92 0.308 76226 0.256
7 2.71 47.22 106.59 501.10 286.18 0.420 73727 0.195
8 2.71 47.26 105.74 429.43 244.83 0.350 72244 0.308
9 2.76 47.22 107.10 366.27 209.18 0.307 70715 0.255
10 2.58 47.22 109.09 573.94 327.78 0.514 68428 0.320
11 2.71 46.56 109.51 567.18 333.04 0.523 69396 0.321
87
12 2.69 46.79 107.40 587.51 341.78 0.522 73259 0.221
13 2.70 47.03 114.56 563.95 324.67 0.489 73579 0.198
14 2.70 47.04 112.74 463.92 266.93 0.354 75840 0.196
15 2.70 47.18 111.19 331.15 189.44 0.234 81580 0.230
16 2.70 46.15 112.69 580.26 346.84 0.537 71132 0.306
17 2.70 46.16 112.44 348.66 208.28 0.393 82415 0.238
18 2.70 46.13 113.25 568.07 339.96 0.498 73413 0.333
Average 2.70 47.12 116.62 461.76 267.37 0.397 74.34 0.26
88
Table A.2: Laboratory static Brazilian test results for meta-sandstone samples
Sample
No.
Density
(g/cm3)
Diameter
(mm)
Length
(mm)
Failure load
(KN)
Static Tensile
Strength (MPa)
1 2.71 47.02 31.01 45.70 19.95
2 2.68 47.15 32.01 42.93 18.11
3 2.69 47.22 32.30 45.89 19.15
4 2.70 47.07 32.80 53.26 21.96
5 2.69 47.21 32.26 45.98 19.22
6 2.69 47.18 33.25 55.67 22.59
7 2.71 47.16 31.94 53.86 22.76
8 2.66 47.09 32.36 47.14 19.69
9 2.73 47.12 31.69 34.03 14.51
10 2.70 47.22 27.09 46.31 23.05
11 2.71 47.26 27.97 37.33 17.98
12 2.71 47.26 29.24 35.74 16.47
13 2.70 46.18 27.24 44.16 22.35
14 2.68 46.61 28.48 45.18 21.67
15 2.69 46.90 29.31 46.97 21.75
89
16 2.69 47.02 31.55 60.06 25.77
17 2.69 47.01 31.24 48.01 20.81
18 2.69 47.20 30.30 54.17 24.11
19 2.70 46.14 32.39 63.02 26.85
20 2.69 46.27 31.85 60.09 25.96
21 2.69 46.15 30.82 59.02 26.42
Average 2.70 46.91 31.21 48.79 21.48
90
Table A.3: Laboratory drop test results for meta-sandstone samples
Sample
No.
Density
(g/cm3)
Diameter
(mm)
Length
(mm)
Failure load
(KN)
Dynamic Tensile
Strength (MPa)
1 2.69 47.04 30.39 51.02 22.72
2 2.68 47.13 32.86 52.72 21.67
3 2.69 47.07 32.43 88.11 36.75
4 2.68 47.08 30.77 58.53 25.72
5 2.69 47.06 30.42 72.29 32.15
6 2.68 47.07 33.68 74.65 29.98
7 2.74 47.17 33.81 62.80 25.07
8 2.69 47.19 32.22 65.27 27.33
9 2.67 47.05 31.94 66.01 27.96
10 2.72 47.24 30.28 51.90 23.10
11 2.70 47.22 31.54 58.81 25.14
12 2.67 47.11 31.02 82.83 36.09
13 2.67 47.03 33.55 65.48 26.42
14 2.68 46.99 29.88 77.69 35.23
15 2.69 47.36 26.54 47.33 23.97
16 2.70 47.36 27.94 54.69 26.31
91
17 2.76 47.32 26.09 58.65 30.24
18 2.71 47.19 30.27 68.00 30.31
19 2.69 47.24 32.10 61.25 25.71
20 2.75 47.08 31.96 61.42 25.98
21 2.75 47.13 32.02 71.82 30.30
22 2.75 47.16 33.41 76.28 30.82
23 2.73 47.22 31.48 75.96 32.53
24 2.71 46.27 28.76 58.94 28.20
25 2.70 46.73 30.13 50.35 22.76
26 2.68 46.86 28.44 56.76 27.12
27 2.71 46.33 31.28 80.35 35.30
28 2.68 46.96 30.41 76.40 34.06
29 2.70 47.04 31.14 77.87 33.84
30 2.69 47.09 31.04 62.68 27.30
31 2.70 47.18 31.08 78.48 34.07
32 2.70 47.08 30.76 60.61 26.64
33 2.67 47.14 30.18 66.58 29.79
34 2.71 47.09 31.73 83.77 35.69
35 2.71 47.06 33.15 87.08 35.54
92
36 2.71 47.10 31.28 86.18 37.24
37 2.70 46.15 32.45 62.43 26.54
38 2.70 46.18 32.38 65.86 28.04
39 2.70 46.12 31.53 62.90 27.54
40 2.71 46.14 31.34 56.95 25.07
41 2.70 46.17 30.64 58.86 26.49
42 2.69 46.13 31.31 92.22 40.65
43 2.68 46.10 33.25 87.19 36.21
44 2.70 46.12 32.31 99.61 42.55
Average 2.70 46.89 31.33 68.54 29.82
93
Appendix B: Bonded Particle Model Genesis Procedure
This appendix presents the procedure for the generation of bonded particle model, which is
quoted as follows by Potyondy and Cundall (2004):
“1. Compact initial assembly: A material vessel consisting of planar frictionless walls is created,
and an assembly of arbitrarily placed particles is generated to fill the vessel. The vessel is a
rectangle bounded by four walls for PFC2D and a rectangular parallelepiped bounded by six
walls for PFC3D. The normal stiffness of the wall is made slightly greater than the average
particle normal stiffness to ensure that the particle-wall overlap remains small. The particle
diameters satisfy a uniform particle size distribution bounded by Dmin and Dmax (minimum
and maximum diameter of the particles). To ensure a reasonably tight initial packing, the
number of particles is determined in such a way that the overall desired porosity in the vessel
is achieved. The particles, at half their final size, are placed randomly such that no two
particles overlap. Then, the particle radii are increased to their final values, and the system is
allowed to rearrange under zero friction, Figure B-1a.
2. Install specified isotropic stress: The radii of all particles are reduced uniformly to achieve a
specified isotropic stress, ơ0 defined as the average of the direct stresses. These stresses are
measured by dividing the average of the total force acting on opposing walls by the area of
the corresponding specimen cross-section. Stresses in the PFC2D models are computed
assuming that each particle is a disk of unit thickness. The magnitude of the locked-in forces
(both tensile and compressive) is comparable to the magnitude of the compressive forces at
the time of bond installation, Figure B-1b.
3. Reduce the number of "floating" particles: An assembly of non-uniform-sized circular or
spherical particles, placed randomly and compacted mechanically, can contain a large
number (perhaps as high as 15%) of "floating" particles that have less than Nf contacts, as
shown in Figure B-1c. Nf is the minimum acceptable number of contacts for each particle in
the assembly which is generally considered as Nf =3. It is desirable to reduce the number of
floating particles so that a denser bond network can be obtained in step 4.
4. Install parallel bonds: Parallel bonds are installed throughout the assembly between all
particles that are in close proximity (with a separation of less than 10-6
time the mean radius
of the two particles), as shown in Figure B-1d.
94
5. Remove from material vessel: The material-genesis procedure is completed by removing the
specimen from the material vessel and allowing the assembly to relax. This is done by deleting
the vessel walls and stepping until static equilibrium is achieved. "
Figure B-1: Material-genesis procedure for a PFC2D model, after Potyondy and Cundall
(2004).