FRACTURE TOUGHNESS BASED MODELS FOR THE

PREDICTION OF POWER CONSUMPTION, PRODUCT SIZE,

AND CAPACITY OF JAW CRUSHERS



JAMES G. DONOVAN



Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and
State University in partial fulfillment of the requirements for the degree of



DOCTOR OF PHILOSOPHY

IN

MINING AND MINERALS ENGINEERING




DR. MARIO G. KARFAKIS
DR. GREG ADEL
DR. NORM DOWLING
DR. TOM NOVAK
MR. HUGH RIMMER
DR. ERIK WESTMAN




JULY 2003
BLACKSBURG, VA


KEYWORDS: Size reduction, Fracture toughness, Jaw crusher, Comminution modeling


Copyright 2003, James G. Donovan
FRACTURE TOUGHNESS BASED MODELS FOR THE PREDICTION OF POWER
CONSUMPTION, PRODUCT SIZE, AND CAPACITY OF JAW CRUSHERS


James G. Donovan


(ABSTRACT)

There is little process control employed at aggregate crushing plants and essentially no
optimization at the primary or jaw crushing stage. Jaw crusher selection is very
dependent on the subjective judgment/experience of individuals, the characterization of
rock material using inadequate and unrepresentative tests, and the desire to limit
secondary breakage, resulting in the conservative selection and operation of jaw crushers.
A method for predicting the power consumption, product size, and volumetric capacity of
jaw crushers based on fracture toughness has been proposed in this study. A new fracture
toughness test, the Edge Notched Disk Wedge Splitting test, has been developed and
verified in order to rapidly assess the fracture toughness of six quarry rocks. A High
Energy Crushing Test system has been used to simulate the operational settings of a jaw
crusher so that comparison of fracture toughness, specific comminution energy, and
breakage distribution could be performed. The results indicate that the specific
comminution energy required to reduce a rock particle to a given size increases with
fracture toughness. The breakage distribution has also been shown to be dependent upon
fracture toughness as long as the elastic modulus is taken into account. Laboratory jaw
crushing experiments show that the capacity of a jaw crusher is dependent upon fracture
toughness and the elastic modulus. Models for the prediction of power consumption,
breakage function/product size, and volumetric capacity have been developed based on
these results. At the experimental level, the models were able to predict the specific
comminution energy to within 1% and t
10
(characteristic crushing parameter) to within
10%. Prediction of the product size distribution produced by a lab-scale jaw crusher, for
four different rocks, was within 5% (in terms of percent passing). The models allow
for the selection of a jaw crusher based on the nature of the rock being broken and the
average amount of size reduction done on the feed material. The models can also be used
to optimize feed and operational settings, as well to determine the product size produced
for a given rock and reduction ratio.




ACKNOWLEDGEMENTS

I want to thank the Department of Mining and Minerals Engineering for allowing me to
continue my graduate studies and conduct this unsupported research. I am grateful for
the financial support, as well as the opportunity to be a part of such a fine department.

I want to recognize the support provided by Luck Stone Corporation, who donated all
seven rock samples in the form of drill core and bulk samples taken from their production
sites. Id also like to thank Bob Stansell for answering all my crushing questions in
regards to Luck Stones operations and in general. And I want to thank Bruce Faison for
accompanying me on my site visits and helping me acquire rock samples.

I want to acknowledge the contributions and involvement of my advisor, Dr. Mario
Karfakis. He provided the motivation for this project but gave me complete liberty in
developing, and executing, my research approach. He provided constant support along
the way and was always available as a sounding board for ideas and suggestions. I feel
fortunate to have been his student for the past six years.

I want to also thank the rest of my committee members, Dr. Adel, Dr. Dowling, Dr.
Novak, and Dr. Westman, for their guidance and support. This work would not be the
same if not for their participation and input. I was lucky to have such an informed and
knowledgeable group of people at my disposal. I want to also thank Hugh Rimmer for
stepping in as a last minute addition to my committee, as well as for making possible the
use of the High Energy Crushing Test system in this study. I am pleased to have
demonstrated how beneficial his donation of the HECT system can be.

Finally, Id like to personally recognize the two people that make all things possible,
and all things bearable, for me: my wife, Erin, and my son, Quinn. To Quinn, you make
me realize everyday just how unimportant the material on the following pages is. To
Erin, I couldnt be given enough pages, enough words, or enough time to properly
express what you mean to me, and what youve done for me the past four years. In the
same time it took me to write this dissertation you have defined me in the two most
important ways, as a husband and a father, and that will always be the first thing I recall
when thinking about this time in our lives.
iii
TABLE OF CONTENTS

PAGE
ABSTRACT
ACKNOWLEDGEMENTS
TABLE OF CONTENTS
LIST OF FIGURES
LIST OF TABLES

CHAPTER 1. INTRODUCTION
1.1 PROJECT OBJECTIVES

CHAPTER 2. LITERATURE REVIEW
2.1 PRIMARY CRUSHING
2.1.1 JAW CRUSHERS
2.1.2 JAW CRUSHER SELECTION
2.1.3 PREDICTING JAW CRUSHER PERFORMANCE
2.1.4 CONCLUSIONS
2.2 PHYSICS OF PARTICLE FRACTURE
2.2.1 PARTICLE BEHAVIOR
2.2.2 GRIFFITHS THEORY
2.2.3 FRACTURED SIZE DISTRIBUTION
2.2.4 CONCLUSIONS
2.3 FRACTURE ENERGY AND SIZE REDUCTION
2.3.1 LAWS OF COMMINUTION
2.3.2 CONCLUSIONS
2.4 SINGLE PARTICLE BREAKAGE ANALYSIS
2.4.1 CONCEPTS OF SINGLE PARTICLE BREAKAGE
2.4.2 TEST METHODS
2.4.3 DATA COLLECTION
2.4.4 CONCLUSIONS
2.5 ROCK FRACTURE MECHANICS AND FRACTURE TOUGHNESS
2.5.1 LINEAR ELASTIC FRACTURE MECHANICS
2.5.2 STRESS INTENSITY FACTOR
2.5.3 MODE I FRACTURE TOUGHNESS TESTING
2.5.4 APPLICATIONS OF ROCK FRACTURE TOUGHNESS
2.5.5 CONCLUSIONS

CHAPTER 3. EXPERIMENTAL PROGRAM
3.1 ROCK SPECIMENS
3.1.1 ROCK PROPERTIES
3.2 FRACTURE TOUGHNESS TESTING
3.2.1 DEVELOPMENT OF THE EDGE NOTCHED DISC WEDGE SPLITTING TEST
3.2.2 EXPERIMENTAL VERIFICATION OF THE END WEDGE SPLITTING TEST
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3.2.3 APPLICATION OF THE END WEDGE SPLITTING TEST
3.3 SINGLE PARTICLE BREAKAGE TESTING
3.3.1 CONCEPTS OF THE HIGH ENERGY CRUSHING TEST
3.3.2 HECT TEST SET-UP AND PROCEDURE
3.4 DATA REDUCTION AND MODEL DEVELOPMENT
3.5 LABORATORY SCALE CRUSHING TESTS
3.5.1 CRUSHING EQUIPMENT
3.5.2 ROCK SAMPLES
3.5.3 TEST PROCEDURE AND DATA ACQUISITION

CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION
4.1 FRACTURE TOUGHNESS
4.2 SINGLE PARTICLE BREAKAGE
4.2.1 PARTICLE STRENGTH
4.2.2 SPECIFIC COMMINUTION ENERGY
4.2.3 BREAKAGE FUNCTION

CHAPTER 5. MODEL DEVELOPMENT AND EXPERIMENTAL VERIFICATION
5.1 MODELS FOR JAW CRUSHER POWER CONSUMPTION AND PRODUCT SIZE
5.1.1 POWER CONSUMPTION
5.1.2 PRODUCT SIZE
5.2 EXPERIMENTAL VERIFICATION OF MODELS
5.2.1 LABORATORY VERIFICATION OF E
C
AND BREAKAGE FUNCTION MODELS
5.2.2 PREDICTION OF PRODUCT SIZE USING A LABORATORY JAW CRUSHER
5.3 MODEL FOR VOLUMETRIC CAPACITY OF A JAW CRUSHER

CHAPTER 6. CONCLUSIONS AND RECOMMENDATIONS
6.1 CONCLUSIONS
6.2 RECOMMENDATIONS

REFERENCES

APPENDIX I. END RESULTS: FRACTURE TOUGHNESS
APPENDIX II. HECT RESULTS: SPECIFIC COMMINUTION ENERGY
APPENDIX III. HECT RESULTS: BREAKAGE SIZE DISTRIBUTION
APPENDIX IV. LABORATORY JAW CRUSHING RESULTS
APPENDIX V. EQUILIBRIUM ANALYSIS OF WEDGE FORCES

VITA

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v
LIST OF FIGURES

PAGE
2.1 Cross-section of Nordberg C Series jaw crusher
2.2 Schematic of crusher dimensions
2.3 Stress distribution in particle under localized compressive loading
2.4 Breakage and classification in a jaw crusher
2.5 Product size distributions based on CSS
2.6 Specimen and loading configuration for compressive strength
2.7 Stress-strain behavior of three different rocks
2.8 Flowchart of classification and breakage process
2.9 Whiten crusher model
2.10 Whiten classification function
2.11 Elliptical hole in an infinite plate
2.12 Fracture caused by compression crushing
2.13 Size distributions occurring due to mechanisms of fracture
2.14 Size ranges applicable to Von Rittinger, Kick, and Bond equations
2.15 Phenomena that effect single particle fracture
2.16 Ultrafast load cell configuration
2.17 Hopkinson pressure bar experimental set-up
2.18 JKRMC drop weight test set-up
2.19 Twin pendulum test set-up
2.20 A typical breakage function and corresponding parameters
2.21 One-parameter family of curves
2.22 Relationship between product size and specific comminution energy
2.23 The three basic modes of crack surface displacement
2.24 Coordinate system for a crack tip
2.25 Development of the FPZ
2.26 Size and shape of the FPZ
2.27 The SB specimen and testing configuration
2.28 The SR specimen and test configuration
2.29 The SCB specimen and test configuration
3.1 Location of rock quarries that provided specimens
3.2 Physiographic provinces of Virginia
3.3 Seven quarry rocks to be tested
3.4 Edge notched disk specimen and loading configuration
3.5 Comparison of independent solutions of F
p
vs. a/D for an END
3.6 Forces acting at the crack mouth
3.7 Wedge configuration
3.8 Test set-up for END wedge test
3.9 Example vertical load vs. calculated CMOD curve
3.10 Fracture toughness vs. a/D
3.11 HECT system
3.12 Displacement profile of HECT actuator
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3.13 Schematic and picture of HECT loading configuration
3.14 Recorded and processed curves from DATA 6000
3.15 Size distribution of samples crushed in laboratory jaw
4.1 Fracture toughness data for all valid specimens
4.2 Correlation between fracture toughness and tensile strength
4.3 Correlation between fracture toughness and elastic modulus
4.4 Correlation between fracture toughness and compressive strength
4.5 Correlation between dynamic tensile strength and tensile strength
4.6 Correlation between dynamic tensile strength and elastic modulus
4.7 Correlation between dynamic tensile strength and compressive strength
4.8 Correlation between dynamic tensile strength and fracture toughness
4.9 Typical BG HECT behavior
4.10 Typical CRS HECT behavior
4.11 Typical CGS HECT behavior
4.12 Typical LD HECT behavior
4.13 Typical SG HECT behavior
4.14 Typical SMB HECT behavior
4.15 Typical TG HECT behavior
4.16 Correlation between E
c
and tensile strength
4.17 Correlation between E
c
and elastic modulus
4.18 Correlation between E
c
and compressive strength
4.19 Correlation between E
c
and fracture toughness
4.20 Correlation between E
c
and dynamic tensile strength
4.21 Breakage functions for each rock at reduction ratio 1
4.22 Breakage functions for each rock at reduction ratio 2
4.23 Breakage functions for Boscobel granite
4.24 Breakage functions for Culpeper gray siltstone
4.25 Breakage functions for Culpeper red siltstone
4.26 Breakage functions for Leesburg diabase
4.27 Breakage functions for Spotsylvania granite
4.28 Breakage functions for Shadwell metabasalt
4.29 Breakage functions for Thornburg granite
4.30 Relationship between t
10
and E
c

4.31 Relationship between t
10
and K
Ic

4.32 Relationship between various t
n
and K
Ic
for reduction ratio 1
4.33 Relationship between various t
n
and K
Ic
for reduction ratio 2
4.34 t
n
vs. t
10
overlapped with Narayanans one-parameter family of curves
5.1 Linear fit of E
c
and K
Ic
data for reduction ratio 1
5.2 Linear fit of E
c
and K
Ic
data for reduction ratio 2
5.3 Change in coefficient m with reduction ratio
5.4 Allometric fit of t
10
versus G for reduction ratio 1
5.5 Allometric fit of t
10
versus G for reduction ratio 2
5.6 Change in coefficient a with reduction ratio
5.7 Change in exponent b with reduction ratio
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5.8 Plot of functions relating large t
n
values

and t
10

5.9 Plot of functions relating small t
n
values

and t
10

5.10 Measured and predicted breakage function for dolomitic limestone
5.11 Predicted and actual product size distributions for Spotsylvania granite
5.12 Predicted and actual product size distributions for Shadwell metabasalt
5.13 Predicted and actual product size distributions for Culpeper siltstone
5.14 Predicted and actual product size distributions for Boscobel granite
5.15 K
3
parameter versus fracture toughness
5.16 K
3
parameter versus strain energy release rate

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viii
LIST OF TABLES

PAGE
2.1 Capacities for various Nordberg C Series jaw crushers
2.2 Material hardness rankings based on compressive strength
3.1 Mechanical properties of tested rocks
3.2 Values of a/D and corresponding value of F
p
from Isida (et al., 1979)
3.3 Values of and from tilt test on hardened steel
3.4 Fracture toughness values from the END test and SCB test
3.5 Average specimen dimensions and coefficient of friction for tested rocks
3.6 Average HECT test specimen dimensions
4.1 Fracture toughness results
4.2 Tensile strength measured in HECT tests
4.3 Specific comminution energy results from the HECT
4.4 Values of t
10
for each rock and reduction ratio
5.1 Strain energy release rate of each rock
5.2 Actual and predicted E
c
and t
10
for dolomitic limestone
5.3 Rock specific volumetric capacities of laboratory jaw crusher
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ix
CHAPTER 1. INTRODUCTION


The size reduction of brittle materials is the most essential mechanical operation within
the raw material processing, i.e. mining, industry. It is also an inefficient, energy
intensive process that consumes billions of kilowatt-hours of electricity per year
(approximately 3-5% of all electricity consumed on the national level (Hofler, 1990;
Duthoit, 2000)). In fact only 1% of the total energy input into size reduction processes is
used in fragmentation and the creation of smaller particles, with most of the energy
manifesting in the form of heat and noise. How has a process so fundamental, and costly,
to the mining industry remained so inefficient? In a large part because the scientific
research required to lay down the theoretical foundations of particle size reduction has
lagged behind the actual achievements of technology, resulting in the design and
operation of crushing equipment based on standards that fail to adequately describe the
entire particle breakage process (Beke, 1964). Since the technology is already in place,
improvements in comminution are dependent upon optimizing the application and
operation of that technology. In fact, the United States National Materials Advisory
Board estimated that improving the energy efficiency of comminution processes, using
practical approaches, could result in energy savings of over 20 billion kilowatt-hours per
year (Napier-Munn et al., 1999).

The failure to optimize the selection and operation of large, costly crushing
equipment affects the entire mining industry. However, the aggregate industry is
particularly burdened by the utilization of inefficient crushers due to the absolute
dependence on the crushing plant to provide a finished product that provides all the
revenue for the mining operation (Svensson and Steer, 1990). Aggregate producers are
reliant on crushers to produce low value materials at high-throughputs that also must
meet stringent quality standards (i.e., particle size and shape for Superpave mix design).
Furthermore, the design of aggregate crushing processing plants requires rugged,
massive, and expensive equipment, and misjudgments in design are difficult, if not
impossible, and expensive to correct (Duthoit, 2000). In todays crushing environment
the need for optimization of primary crusher selection and operation to meet the
economics of quarrying requirements has become imperative (DeDiemar, 1990). The
benefits of optimization include reduced capital costs, reduced unit operating costs,
increased throughput, and improved performance of downstream processes as a result of
improved feed size specification (Napier-Munn et al., 1999).

Much work has been done in the area of size reduction with regards to developing a
theory, or criterion, that can be used to select and evaluate crushing equipment. Various
theories have been proposed, none of which is completely satisfactory. Von Rittinger,
Kick, and Bond proposed theories that constitute the three laws of comminution, all of
which primarily demonstrate a relationship between the size reduction ratio of a material
and the necessary energy input. The inherent problem with each of these laws is that
they are empirically based laws that only fit experimental data over a limited range of
1
variables and only in certain cases (Choi, 1982). Furthermore, the relevance of these
laws to primary crushing equipment processing large particle feed sizes is questionable.
Eloranta (1997) calculated the expected power consumption of a crusher using Bonds
theory and found that the actual power consumption was 240% higher than the
expected consumption. In fact, crusher manufacturers and design engineers rely on a
host of other methods to size and select crushing equipment for aggregate operations.
Unfortunately these methods often lend themselves to the subjective judgment of an
individual, resulting in the conservative over-design of crushing plant equipment
(Bearman et al., 1990).

The importance of Von Rittinger, Kick, and Bonds work is that it indicates some
relationship between the energy required to decrease the size of a particle and the
resultant size of the broken particle. Bonds work in particular highlights the importance
of selecting and evaluating crushing equipment, and determining power requirements,
based on product size and some measure of a materials resistance to fracture. Over the
last three decades more research has focused on the physics of particle fracture during the
crushing process and the material characteristics related to fragmentation. Advancements
in comminution theory have been made during this time by conducting single particle
breakage tests, the goal of which is to relate the pattern of breakage, and the resultant
fragment size distribution, with the nature of the material broken (Arbiter et al., 1969).
Single particle breakage studies are now used to develop mathematical models, or
functions, that describe the size reduction process of different materials. The trend in
single particle breakage analysis now is to further relate the fracture energy and product
size distribution to some physical property of the material. The ultimate purpose is to be
able to assess the fracture properties of a material, and the requirements of any device
used to break that material, using a single, easily determined physical property.

There is evidence that a link may exist between characteristic rock properties and the
energy consumption, and performance, of primary crushing equipment. Berry at al.
(1984) has related a number of common rock strength parameters to the performance of a
laboratory scale crusher. Bearman et al. (1991) performed tests that lead to the
development of empirical relationships between a number of rock strength parameters
and the power consumption, and product size, of a cone crusher. It has been suggested
that a materials fracture properties should be characterized in terms of particle strength,
specific breakage energy, and the breakage fragment size distribution of the material
(Bourgeois et al., 1992). In this case the strength is measured in terms of the tensile
strength because particles of brittle materials break in tension under compressive loads
due to the pre-existence of natural flaws within the material. However, the tensile
strength does not fully describe the fracture process, it is only a measure of a particles
resistance to an applied load. It does not account for the inherent flaws present in brittle
materials, or the magnitude of the stresses around these flaws when the material is
loaded, and their effect on the fracture process. Thus, the particle strength should be
replaced by another, more descriptive measure of a materials ability to withstand
fracture.
2

Fracture toughness is an intrinsic material property expressing a materials resistance
to crack propagation, and it is a measure of the energy required to create a new surface in
a material. More specifically, fracture toughness is a critical value of the stress intensity
factor, a quantity that characterizes the severity of the crack situation within a material as
affected by crack size, stresses near the crack tip, and the geometry of the crack, material
specimen, and loading configuration (Dowling, 1999). Given its sound theoretical
foundation, evolving from the study of fracture mechanics and Griffiths theory of
fracture, fracture toughness may offer a fundamental basis for describing size reduction
processes. It is already used as an index for rock fragmentation processes such as tunnel
boring and scale model blasting, as a key element in modeling rock cutting and blasting,
and in the stability analysis of rock structures (Whittaker et al., 1992). Furthermore,
Bearman (et al, 1989) has shown that a strong correlation exists between the fracture
toughness of a material and the power consumption of a laboratory crusher used to crush
the material, indicating that fracture toughness may have practical application in the
evaluation of crushing equipment.

The major drawback in using fracture toughness as a parameter for crushing
equipment evaluation is that the tests required to measure the property are difficult and
expensive to perform, particularly in the case of brittle materials such as rock. The
complexity of existing tests is an immediate problem when considering the application of
fracture toughness to gauge size reduction processes (Bearman, 1998). This includes the
standard testing techniques established by the International Society of Rock Mechanics
(ISRM), tests that, although employing core-based specimens, require a chevron-notch to
be machined into each specimen, a tedious and difficult process, especially for a large
number of specimens. Thus, another method for fracture toughness testing of rocks is
necessary, one that will give a representative toughness value and yet be simple,
requiring neither pre-cracking, nor crack-length, displacement measurements, or
sophisticated evaluation techniques (Ouchterlony, 1989).

Based on the premise that fracture toughness fully characterizes a rocks resistance to
fracture, it is proposed to examine the potential of fracture toughness to function as a
predictive means for rock crushability, and to develop fracture toughness based models
for the selection and optimization of primary jaw crushers. The models will be
developed by investigating the relationships among fracture toughness, breakage energy,
and fragment size distributions of various aggregates. Fracture toughness will be
measured using a newly developed fracture toughness test, and the energy and particle
size distributions will be measured using a unique single particle breakage system. The
ability of the laboratory derived models to predict the power draw and product size of jaw
crushers will be examined using a lab-scale jaw crusher, and if possible, with the working
conditions of large-scale primary crushers operated by Luck Stone Corporation.



3
1.1 PROJECT OBJECTIVES

Black and Hoek (1967) outlined the general methods of applied science used when
approaching problems involving rock mechanics. Following their outline the objectives
of this project are:

1. Observe the natural phenomena in question as they occur and if possible quantify
the observations: Primary crushing equipment used in the aggregate industry is
inefficient and over-designed. The proper selection and optimization of primary
crushing equipment should be based on the energy required for rock breakage, the
desired product size, and the desired production capacity, all of which are
dependent upon a proper physical characterization of the rock being crushed.

2. Propose a model or theory to explain the phenomena observed: The
fragmentation of rock in any size reduction process relies on the propagation of
internal flaws within the rock. The resistance of a rock to crack propagation can
be quantified using fracture toughness; it is a measure of the energy required to
cause extension of flaws present within the rock. The power consumption,
product size distribution, and capacity of a jaw crusher are dependent on the
distribution of flaws within the rock being broken. It follows that fracture
toughness can be used to optimize the selection and performance of a jaw crusher.

3. Test and develop the model through laboratory experimentation: The fracture
toughness of various rock types will be determined using a newly developed
procedure that utilizes a core-based specimen and allows for the rapid estimation
of fracture toughness. The High Energy Crush Test (HECT) system will be used
to determine the specific comminution energy and breakage function of the same
rock types. Regression analysis of the data will be used to develop models for the
prediction of power consumption and breakage function based on fracture
toughness. Laboratory scale jaw crusher experiments will be used to redefine the
current model for volumetric capacity taking into account fracture toughness.

4. Use the model to predict behavior in similar circumstances: Dolomitic limestone
will be tested in the HECT system in order to test the applicability of the specific
comminution energy and breakage function models. A laboratory scale jaw
crusher will be used to test the ability of the breakage function model to predict
the product size for a given set of operational settings.

5. Insure by measured experiment or observation that the predictions made are
valid, or that deviations from the behavior can be accounted for: In addition to
the laboratory scale crusher, jaw crushers currently in use at Luck Stone
Corporations quarrying operations will be evaluated. The effects of specimen
size, crusher throughput, feed conditions, and effects of blasting will be
considered
4
CHAPTER 2. LITERATURE REVIEW


2.1 PRIMARY CRUSHING

The first stage of rock breakage in any mining application is blasting. The purpose of
blasting is to produce rock fragments small enough to be accepted by the processing
plant. Blasted rock is delivered to the processing plant and subjected to the first stage
of mechanical breakage, primary crushing. Primary crushers are large, rugged machines
capable of handling large quantities of material and are used to reduce the size of blasted
rock to sizes suitable for feeding secondary crushers. They typically handle feed sizes up
to 1.5 meters with reduction ratios (ratio of particle feed size to product size) ranging
from 3 to 10 (King, 2001).

Crushing in general is an energy intensive process. Primary crushing in particular
consumes large amounts of energy due to the significant amount of size reduction taking
place. In addition to the amount of size reduction, the energy required for breakage in
crushing applications is dependent upon the physical properties of the material and the
quantity of material being crushed. The rate of energy input is dependent upon the type
of crushing machine used since the application of the crushing force changes with
machine type.

Primary crushers apply breakage forces by means of compression or impact. Jaw and
gyratory crushers are the most common types of primary compression crushers. Each
applies a compressive force to rock particles as they come in contact with the crushing
surfaces. The force is applied slowly (in comparison to impact machines) resulting in
abrasion and cleavage fracture. Impact crushers apply a high-speed impact force to rock
particles using hammers or blow bars. The rate of energy input is much higher causing
particles to shatter. Impact crushers can achieve higher reduction ratios than jaws and
gyratorys but are limited by high rates of abrasive wear and thus are restricted to
somewhat softer rocks (Duthoit, 2000). In the aggregate industry, for tonnage outputs
less than 1000 metric tons per hour, jaw crushers are the dominant type of primary
crusher.

2.1.1 JAW CRUSHERS

Jaw crushers have been around for almost 175 years. All jaw crushers are distinguished
by the presence of two plates, one of which is fixed and one that swings open and then
closes, concurrently trapping and crushing material between the two surfaces. There are
three types of jaw crushers: Blake, Dodge, and Universal. They are classified according
to the location of the pivot point of the swinging jaw. The most common type of jaw
crusher today is the Blake crusher, patented by Eli Whitney Blake in 1858 (Weiss, 1985).
The Blake crusher pivots at the top and comes in two forms: the double toggle and the
single toggle. The single toggle is taking over most new applications due to lower cost
5
and higher capacity (Stansell, 2003). Figure 2.1 is a cross-section through a Nordberg C
Series single toggle jaw crusher.




Figure 2.1 Cross-section of Nordberg C Series jaw crusher
(From Metso-Minerals 2003)































From Figure 2.1 it can be seen that the swing jaw is suspended from the eccentric
shaft, allowing it to move vertically as the eccentric shaft rotates. The elliptical motion
of the swing jaw assists in pushing rock through the crushing chamber and results in the
single-toggle machines higher capacity (Wills, 1992). The toggle mechanism moves the
swing jaw towards the fixed jaw, and along with the eccentric supplies the force required
for crushing. The flywheel is responsible for the momentum required to maintain a
constant speed during the crusher cycle.
6
Dimensions and operating parameters
When considering the jaw crusher of Figure 2.1, there are variables of the feed that define
the important machine dimensions (Napier-Munn et al., 1999). The feed particle sizes of
interest are:
The size of particle that enters the crusher
The size of particle that can be nipped
The size of particle that can fall through the chamber at any time
The size of particle that can fall through the chamber when the jaws are open as
wide as possible.
The dimensions defined by those particle sizes are (Fig 2.2):
The gape - the distance between the jaws at the feed opening
The closed side set (CSS) - the minimum opening between the jaws during the
crushing cycle (minimum discharge aperture)
The open side set (OSS) the maximum discharge aperture
The throw the stroke of the swing jaw and the difference between OSS and
CSS.

Gape
OSS
CSS
Throw
Figure 2.2 Schematic of crusher dimensions





















Jaw crushers are rated, or designated, according to the maximum dimensions of their
feed opening, defined by the gape and the width of the plates. Feed openings can vary
from 250 mm by 500 mm (10 in 20 in) to 1524 mm by 2032 mm (60 in 80 in). It is
common practice to limit the largest particle of rock entering the jaw to be no more than
80% of the rated gape (Weiss, 1985).

7
The size distribution of the product is influenced mainly by the open and closed side
settings of the crusher. The discharge opening of a jaw crusher is commonly listed in
terms of the closed side set. Closed side sets can range from 19 mm (0.75 in) up to 711
mm (28 in). Crusher sets can be changed on-site, either manually or automatically,
usually in order to compensate for jaw liner wear (King, 2001). The capacity of a jaw
crusher is function of the feed size and the settings of the crusher. Crusher manufacturers
typically produce capacity tables for their various sized machines in terms of the closed
side set (Table 2.1).

Table 2.1 Capacities for various Nordberg C Series jaw crushers
(After Metso Minerals, 2003)
CSS
mm
C63
440 630
C100
760 1000
C110
850 1100
C140
1070 1400
C160
1200 1600
C200
1500 2000
40 40 Capacities in metric tons per hour
50 55
60 65
70 80 150 190
80 95 170 210
90 110 190 235
100 120 215 255
125 265 310 385
150 315 370 455 520
175 370 425 520 595 760
200 420 480 590 675 855
225 655 750 945
250 725 825 1040
275 900 1130
300 980 1225

The throw of a jaw crusher is dependent upon the type of crushing required. For
coarse crushing a long stroke is used and for fine crushing a short stroke is employed,
resulting in a range from 9.5 mm (0.375 in) up to 50 mm (2 in) (Weiss, 1985).

The speed of the crusher is another important operating parameter (expressed in terms
of the flywheel rpm). Single toggle jaw crushers can range in speed from 200 rpm to 350
rpm, and speeds do not typically fall below the 200 rpm threshold. The crusher speed
decreases with increasing machine size and most large single toggle jaw crusher run at
speeds near 225 rpm.

Crushing process
It was previously noted that jaw crushers are compression machines and that under the
action of the eccentric shaft and the toggle, very powerful crushing forces are generated.
In order to fracture a particle the crushing forces must be high enough to exceed the
fracture strength of the particle. When a particle is nipped between the plates of a jaw
8
crusher tensile stresses are induced in the particle. Thus, the compressive force applied
by the jaw plates causes the rock particle to fail in tension (Fig 2.3).

Figure 2.3 Stress distribution in particle under localized compressive loading
(After Oka and Majima, 1970)

After a particle is nipped and fails in tension, the resulting fragments drop down to
new positions within the crushing chamber before being nipped again. Particles continue
to drop down and are either repeatedly nipped by the jaws or pass through the discharge
opening (when small enough). This is referred to as arrested crushing, where crushing is
by the jaws only. The increasing stroke of the swing jaw at the discharge end allows
material to leave at a rate sufficient enough to leave space for the particles above,
preventing choking of the crusher. Since particles smaller than the discharge opening are
free to pass through the crushing chamber at any time, the breakage process within the
crusher operates simultaneously with a classification process (Fig 2.4).
Figure 2.4 Breakage and classification in a jaw crusher
(After Lynch, 1977 and Napier-Munn et al., 1999)
9
2.1.2 JAW CRUSHER SELECTION

An operators control over the crushing process is mainly limited to the selection of the
proper crushing equipment. The main factors that influence the selection of aggregate
primary jaw crushers are (Duthoit, 2000; Weiss, 1985; Metso Minerals 2003a):
The petrographic nature of the rock
The abrasion index of the aggregate
The mechanical strength of the rock
The work index, or impact crushability, of the rock
The brittleness of the rock
The flakiness of the rock
The feed size
The desired product size (or reduction ratio)
The throughput to be crushed (tons per hour)
The capital cost
The long term operating costs.

The petrographic nature of the rock and its resistance to crushing forces
(characterized by strength, work index, etc.) are used first to determine what type of
crusher is needed and then in part to determine the power requirement of the crusher.
When a jaw crusher is selected outright, the physical properties of the rock can be used to
determine the mechanical features of the jaw crusher, but for the most part they are used
along with the feed size and desired throughput to determine the size of the jaw crusher
and its power requirement.

Usually the feed size range or the desired throughput dictates the selection of specific
jaw crusher. The current trend is to go to the largest feed opening that is economically
feasible in order to avoid as much secondary breakage as possible (Stansell, 2003). Once
a crusher is selected it can be determined whether or not the crusher will provide the
desired reduction ratio or product size distribution. As was noted previously the size of a
jaw crushers discharge opening controls the product size distribution. Thus, determining
whether a specific jaw can produce the desired product is dependent upon whether or not
that crusher can have its discharge opening set to the required dimension.

Based on the previous discussion, crusher selection can be simplified down to a
process of employing screen analysis graphs (Fig 2.5) and capacity charts (Table 2.1).
Metso Minerals has developed a computer program to aid in the selection of its Nordberg
C Series jaw crushers. The Bruno program uses input information regarding rock type,
feed gradation, and the desired crusher to simulate expected capacities and product size
distribution curves (Metso Minerals, 2003). Wagner (1990) has proposed a crusher
efficiency index number based on crusher manufacturers published data and the
crushability factor of the rock. However graphs, charts, and other data are usually based
on a standard rock type and although deviations from the standard rock can be accounted
for, there is a level of uncertainty due to the inhomogeneous nature of rock (Bearman et
10
al., 1990). In order to properly characterize the nature of the rock to be crushed various
laboratory tests are usually performed.

Figure 2.5 Product size distributions based on CSS
(From Metso Minerals, 2003)

Rock material classification
Crusher manufacturers perform tests on rock material in order to rank rocks based on
their mechanical properties and to help aid in the determination of crusher power draw.
The main properties used are listed on page 10 but the following discussion focuses on
the most utilized for jaw crushers, compressive strength and the work index.

Compressive Strength
The compressive strength is typically used to rank a rock materials hardness. It is also
sometimes used to determine a materials homogeneity and in the selection of the type of
crusher used (i.e, jaw or gyratory).

Testing for compressive strength follows the standard procedure set forth by the
American Society for Testing and Materials (ASTM D2938). A cylindrical specimen
with a length to diameter ratio of at least 2 is loaded axially (Figure 2.6). The peak load
required to cause failure is then used to determine the compressive strength of the rock
specimen. A number of tests are performed in order to obtain a representative average
value used to rank the material. However, adjustments to the ranking value are
sometimes performed using a ratio of the highest measured value to the average value
(Nordberg, 1995). When the ratio is greater than 175%, the rock is considered to have a
higher than normal variation in its hardness. In this instance crusher manufacturers tend
to the conservative side and use the maximum value to rank the material. Table 2.2 lists
rankings based on compressive strength values.
11

P
P
Figure 2.6 Specimen and loading configuration for compressive strength
L
= P/r
2

= L/L
Table 2.2 Material hardness rankings based on compressive strength
Compressive Strength Ranking
Less than 69 MPa Material is soft. Can be processed by all equipment at low power
consumption. Equipment will be volume limited.
69 to 138 MPa Material is average. Can usually be processed by all equipment.
Can consume full power of equipment.
138 to 276 MPa Material is hard. Require heavy duty equipment and full power
draw.
Greater than 276 MPa Material is extremely hard.

Using compressive strengths to rank materials and/or aid in the selection and design
of jaw crushers has its limitations. The test itself is not representative of the breakage
process occurring inside an actual jaw crusher. As discussed in section 2.1.1, nipping of
a particle between the jaw plates induces tensile stresses within the particle. Thus the
particle fails in tension, a failure process that does not occur during an unconfined
compression test where shear failure dominates. Furthermore, tensile stresses in rocks
tend to propagate, or open, inherent flaws in the material, promoting failure, while
compressive stresses result in crack closure.

In addition to not representing the breakage process occurring within a jaw crusher,
compressive strength alone is not adequate in ranking materials. Peak loads do not fully
describe a rock materials stress-strain behavior, an important aspect when considering
the amount of energy required to crush or break a material. This can be seen in Figure
2.7. Three rock types tested in unconfined compression are compared via their stress-
strain curves. Rock A and Rock B have equal unconfined compressive strengths and
12
rank as hard materials according to Table 2.2. However, Rock B undergoes more
deformation when loaded to failure and thus more work, or energy, is required to break it
(where work is force time distance). The power draw of a jaw crusher used to reduce the
size of Rock A cannot be expected to match that of one crushing Rock B. This point is
further elucidated by Rock C; ranked soft and able to be processed at low power
consumptions (based on compressive strength) it would require full power draw due to its
ability to tolerate a large amount of deformation before fracturing.
0.000 0.004 0.008 0.012 0.016 0.020 0.024 0.028 0.032 0.036 0.040
0
20
40
60
80
100
120
140
160
180
200
Rock C
C
o
= 68 MPa
Energy = 182 J
Rock B
C
o
= 176 MPa
Energy = 183 J
Rock A
C
o
= 176 MPa
Energy = 92 J
S
t
r
e
s
s
,

M
P
a
Strain
Figure 2.7 Stress-strain behavior of three different rocks

Work Index and the crushability test
The work index was developed by Fred Bond (1952) in order to provide a method for
sizing crushing equipment, specify motor sizes, and determine the energy requirements of
the equipment. The Bond crushability test, sometimes referred to as the impact
crushability test, is used to determine the work index.

A particle of rock (representative of what is seen in the actual crushing process)
approximately 75 mm by 50 mm (3 in 2 in) is mounted between two opposing 13.6 kg
(6 lb) hammers that are used to strike the particle simultaneously. The hammers are
repeatedly dropped from increasing heights until the particle breaks. The impact
crushing strength (ICS) is the height from which the hammers were dropped times their
weight divided by the thickness of the particle. An average value is obtained by testing
13
between 10 and 15 particles, and the work index (in kilowatt-hours per metric ton) is
determined from the following formula:
53.49
ICS
WI
SG
= [2.1]
where, ICS is the impact crushing strength is in Joules per millimeter
SG is the specific gravity of the rock.

The impact crushability test is functional and its greatest advantage is that it can be
used to differentiate between materials previously ranked in the same category
according to strength test results. It also includes an energy measurement and a crack
length measurement, essentially tying the energy input to the amount of size reduction,
or the area of new surfaces created. However, only the maximum drop height is used to
determine the ICS although every previous drop contributed to the fracture event and, in
the same vane, it is assumed that the final drop propagated a crack equal to the specimen
thickness in fact each previous drop was contributing to crack growth and fracture
damage. This tends to underestimate the ICS, as more energy is required to cause
complete fracture than is indicated by the final drop height.

2.1.3 PREDICTING JAW CRUSHER PERFORMANCE

The performance of a jaw crusher is most aptly defined using the product size, the
capacity or throughput, and the power consumption. The main object of size reduction in
the aggregate industry is to generate a well-shaped product within a specified size range
with a minimum of fines. It is also desirable to maintain target production rates and to
crush the rock material as efficiently as possible.

There are various methods for the prediction of crusher performance. As noted
previously in the discussion of crusher selection, technical literature provided by crusher
manufacturers can be used to determine capacities and product size. Adjustments can be
made to account for physical variations in feed materials using the aforementioned
laboratory tests. When the actual performance of a jaw crusher is difficult to determine
or unclear to the manufacturer, laboratory tests using small-scale jaw crushers are
sometimes conducted in order to predict the performance of full capacity machines. The
normal procedure is to test crush a representative sample of the feed material and
determine the product size distribution. Comparison of the product distribution and
associated properties of the feed material to other materials is used to estimate overall
crushing performance, power requirements, and to detect characteristics that may require
special design considerations. These tests provide an opportunity to evaluate the effects
of various crusher settings, speeds, feeding methods, and the physical characteristics of
the material (Pennsylvania Crusher Corporation, 2003).

In addition to charts, graphs, and lab-scale crushing tests, jaw crushers can also be
evaluated using mathematical modeling techniques. Mathematical techniques are usually
derived from laboratory data and can be used to predict crusher performance and possibly
14
improve selection and design procedures (Rimmer et al., 1986). Models/equations are
typically used to determine the product size, power draw, and capacity of crushers. The
advantage of models is that they reduce complex operations to a few numbers or
parameters and can provide guidance to improved performance and decision making
(Napier-Munn et al., 1999).

Modeling of product size
Section 2.1.1 covered the crushing process that occurs as feed material passes between
the plates of a jaw crusher. It was noted that the breakage process within the crusher
operates simultaneously with a classification process. Feed material entering the crusher
is first classified, with particles smaller than the CSS of the crusher avoiding breakage
and moving on as product. Particles larger than the OSS of the crusher report to
breakage and will undoubtedly be broken, while particles in between the CSS and OSS
report to breakage but with a probability of being broken that decreases with size. This
process continues with each cycle of the swing jaw and is illustrated as a flowchart in
Figure 2.8.
Figure 2.8 Flowchart of classification and breakage process
(After Napier-Munn et al., 1999)
Feed
Process continues
Product
Product
Breakage Classification
Breakage Classification
Considering the classification and breakage of particles in a jaw crusher as a closed
circuit process reduces Figure 2.8 to the flow path shown in Figure 2.9, a schematic of
the Whiten crusher model (Whiten, 1972). Mass balance equations written about each
node of Figure 2.9 describe the repetitive process of classification and breakage and are
expressed as (Kojovic et al, 1997):
x f Bx = + [2.2]
x p Cx = + [2.3]
where, x is a vector representing the amount in each size fraction entering the crusher
f is the feed size distribution vector
p is the product size distribution vector
C is the classification (diagonal) matrix
15
x
Breakage Function, B
Classification Function, C
Bx
Cx
f p
Figure 2.9 Whiten crusher model
B is the breakage distribution (lower triangular) matrix.

The classification matrix describes the proportion of particles in each size interval
entering the crushing zone. The breakage distribution matrix gives the relative
distribution of each size fraction after a breakage event. Equations 2.2 and 2.3 can be
combined resulting in the Whiten crusher model equation (Whiten, 1972):
( )
1
( ) p C BC

f = [2.4]
where, is the unit matrix. Equation 2.4 can be used to determine the product size of a
jaw crusher given the feed size, classification function, and the breakage function.

The classification function has, in some respect, been defined in the discussion of the
crushing process. The classification function is the probability of a particle being
selected for breakage, which of course is dependent upon particle size. For example, a
particle smaller than the CSS of the crusher probably will not be broken, or has only a
small probability of being selected for breakage. Whiten used the following set of
functions to describe the classification function (Whiten, 1972a):

( )
( )
( )
3
0 for 1
2
1 for 1 2
2 1
1 for 2
K
C x x K
K x
C x K x K
K K
C x x K
= <
| |
= < <
|

\ .
= >
[2.5]
K1 is the particle size below which all particles will by-pass breakage and go directly
to the product, and K2 is the size above which all particles will be broken. These two
parameters are expected to be functions of the crusher sets, K1 being dependent upon
CSS and K2 upon OSS. As an exponent K3 describes the shape of the classification
function for particles between the K1 and K2 sizes (Figure 2.10).

Research has shown that KI and K2 are, in addition to closed and open side set,
functions of feed size, throughput, crusher throw, and plate liner characteristics with
exact relationships being found empirically using actual operating conditions/parameters
(Napier-Munn et al., 1999; Whiten 1984; Karra, 1982; Andersen, 1988; Andersen and
Napier-Munn, 1990). The same work has shown that K3 remains fairly constant among
16
various instances with a value around 2.3. An initial estimate of the classification
function sets K1 equal to the CSS, K2 to the largest particle in the product (should be
close to OSS), and K3 equal to 2.3.

The breakage function is determined from single particle breakage testing (section
2.4). It is a size distribution dependent upon the nature of the material broken and to
some extent the operating conditions of a crusher. Within a jaw crusher, a large number
of single particle breakage events are occurring simultaneously, with a portion of the
products of breakage undergoing subsequent crushing. The breakage distribution
mathematically describes this process as it considers breakage in terms of a series of
single particle fracture events (Kelly and Spottiswood, 1982). A more detailed discussion
of the breakage function can be found in section 2.4.3.

K2 K1
0
P
r
o
b
a
b
i
l
t
y

o
f

b
r
e
a
k
a
g
e
,

C
(
x
)
Particle size, x
1.0
Figure 2.10 Whiten classification function
K3

Csoke (et al., 1996) developed an empirical method for determining the product size
of preliminary crushers, i.e. jaw crushers and gyratory crushers. The product size
resulting from breakage of the material larger than the CSS of the crusher can be modeled
using the following function:
( )
max
m
r
P d
r
| |
=
|
\ .
[2.6]
with,
17

max
p
pMAX
d
r
Gap
d
r
Gap
=
=

where, m is exponent describing the product size distribution
d
p
is the particle size
d
pMAX
is the largest size in the product (OSS)
the Gap is the CSS of the crusher.
Based on operating data found in the literature Equation 2.6 fits the product size of a jaw
crusher with r
max
equal to 1.223 and m of 0.842.

King (2001) outlined a model for the prediction of product size based on data from a
crusher manufacturer. The main concept is that the size distribution of the product is
characterized by the particle size relative to the OSS and by the product type, P
t
, which is
the fraction of the product smaller than the OSS. P
t
is related to the nature of the material
and is distinguished by the work index and a qualitative description of the material. The
size distribution given by this model is determined from the following equations:

( )
( )
1.5
0.85
1 for 0.5
1 for 0.5
u
r
K
r
KL
P d e r
P d e r
| |
| |
|

|
|
\ .
\ .
| |
| |
|
|
|
\ .
\ .
= >
= <
[2.7]
with,

1.5
0.67
1.18
0.5
1
ln
1
1
0.5 ln
1
1
u
i
i
u
t
l
b
K
b
D
r
OSS
K
P
K
P
P e

| |
| |
|

|
|
\ .
\ .
=
( | |
=
( |

\ .
( | |
=
( |

\ .
=

Equation 2.7 should only be used when the size at which half of the feed material passes
is greater then the OSS.

Modeling of power draw
The most popular method of determining the power draw or consumption of size
reduction equipment is Bonds method. Bonds theory states that the work input is
proportional to the new crack tip length produced in particle breakage and is equal to the
18
work represented by the product minus the work represented by the feed (Bond, 1952).
Based on this theory the equation for determining the work input (kWh/metric ton) is:

80 80
1 1
10 W WI
P F
| |
=

\ .
|
|
[2.8]
where, WI is the work index in kWh/metric ton (see section 2.1.2)
P
80
is the size at which 80% of the product passes
F
80
is the size at which 80% of the feed passes.
Using Equation 2.6 the power draw of a size reduction machine can be found for a given
feed throughput from:
P W T = [2.9]
where T is the throughput in metric tons per hour.

Although Bonds equation applies reasonably well to grinding conditions (i.e., ball
mills and rod mills) its relevance to primary crushing equipment processing large feed
sizes is questionable. Bonds law is an empirical law that only fits experimental data
over a limited range of variables and only in certain cases (Choi, 1982). It can be
corrected for other operating conditions but even with this correction it has been shown
that the power consumption of a jaw crusher can be as much as 240% greater than the
expected consumption based on Bonds equation (Eloranta, 1997). Rose and English
have suggested a relationship for power consumption based on Bonds work index but
take into account the material density and machine characteristics, i.e., gape, throw, CSS,
etc. (Lowrison, 1974).

A more recent approach to predicting power consumption has focused on the power
required by laboratory breakage devices to achieve the same size reduction as seen in
actual crushers (Andersen and Napier-Munn, 1988; Morrell et al., 1992). The power
required by the lab device, in this case the twin pendulum or the drop-weight (see section
2.4.2), is related to the actual power drawn by a crusher using the following expression:

c p
P AP P
n
= + [2.10]
where, P
c
is the power drawn by the crusher under load in kW
P
p
is the pendulum power in kW
P
n
is the power drawn by the crusher under no load in kW
A is scaling factor specific to a crusher (from regression analysis).
The process for determining the pendulum power is based on the classification and
breakage functions described previously and will not be covered here. It can be
expressed as the total energy required to reduce the crusher feed size to the product size
as if all the reduction took place in the laboratory device (Napier-Munn et al., 1999).

Estimating capacity
The capacity of a jaw crusher is dependent upon the machine characteristics, the feed,
and the nature of the rock material. The volumetric capacity of a jaw crusher can be
estimated from the following equations (Sastri, 1994):
19

( )
( )
( )
2
60 0.5 at low speeds
450
60 0.5 at high speeds
DT
V Nw CSS T
G CSS T
g
V Nw CSS T
N
| |
= +
|
|
+
\ .
| |
= +
|
\ .
[2.11]
where, N is the speed of the crusher in rpm
w is the width of the jaws in m
CSS is the closed side set in m
T is the throw in m
D is the vertical depth between the jaws in m
G is the crusher gape in m
g is gravitational acceleration in m/s
2
.
Equation 2.11 gives volumetric capacity under ideal conditions. At low crusher speeds
the material falls down the chamber due to gravity flow, and the distance of the fall is
dependent on the geometry of the machine. At higher speeds there is less time between
strokes/revolutions and movement of material down the chamber is constricted and
controlled by the speed of the machine (Hersam, 1923).

In order to determine the actual capacity of a jaw crusher, Equation 2.11 needs to be
corrected by accounting for the feed size, the degree of compaction resulting from
vibration, and the nature of the material. Equation 2.12 gives the estimated capacity
using these parameters (Sastri, 1994):

( )
( )
( )
1 2 3
1 2 3 2
60 0.5 at low speeds
450
60 0.5 at high speeds
DT
V Nw CSS T K K K
G CSS T
g
V Nw CSS T K K K
N
| |
= +
|
|
+
\ .
| |
= +
|
\ .
[2.12]
with,

2.5
1
6.5
2
0.85
1.92 10
avg
T
G
F
K
G
K

| |
=
|
\ .
=
[2.13]
where, F
avg
is the average feed size in m
K
3
is a parameter related to the nature of the material.
The parameter K
3
does not have a suggested value but is considered to increase with
increasing hardness or toughness.

2.1.4 CONCLUSIONS

Jaw crushers are large, heavy-duty machines capable of crushing large quantities of
tough, abrasive materials. They are typically employed as primary crushers within
aggregate processing plants. Jaw crushers are most commonly defined by gape size,
20
open and closed side sets, and capacity. There are a wide range of jaw crusher sizes for
varying production and product size requirements.

The plates of a jaw crusher are used to apply compressive forces that induce tensile
stresses within particles, causing fracture. Particles are repeatedly nipped until they pass
through the crushing chamber. The breakage process occurring between the jaw plates
acts simultaneously with a classification process. The classification process defines
whether or not a particle will be subjected to crushing and is dependent upon the settings
of the crusher and particle size.

The selection of a jaw crusher for application in the aggregate industry is primarily
based on technical literature provided by crusher manufacturers, experience, and cost.
Charts and graphs provide data on electric power requirements, crusher size (gape), as
well as expected capacities for a given material and closed side set. These charts and
graphs have also been incorporated into computer programs to aid in crusher selection.
In order to account for material variations manufacturers rely on a suite of laboratory
tests. The uniaxial compression and Bond impact crushability test are two of the most
common tests used to rank materials relative to their hardness/crushability. Each of these
tests has limitations and neither adequately describes a materials resistance to fracture.
Although jaw crushers are used extensively, the lack of understanding relative to their
operational characteristics, as well as a reliance on an inappropriate, single set of material
properties, makes selection of a proper machine difficult. Jaw crusher selection is also
heavily influenced by the subjective judgment/experience of individuals, which can result
in the conservative selection and operation of jaw crushers.

The prediction of crusher performance is typically concerned with the size
distribution of the product exiting the crusher, the machines power draw, and the
capacity. The capacity of a jaw crusher is dependent upon the operating characteristics of
the machine and influenced by the feed size and the nature of the material. It can be
determined fairly easily but there has been difficulty in accounting for the variable nature
of rock materials. Bonds theory or equation has long been used to determine the power
draw and motor requirements of size reduction machines. However, its relevance to jaw
crushers handling large feed sizes has been questioned. The recent trend has been to
calculate power draw based on laboratory crushing tests. The modeling of crusher
performance has focused mainly on the prediction of the product size distribution. The
product size distribution of a crusher can be determined using the Whiten model given
the feed size, the classification function, and the breakage function. The classification
function is dependent upon the operating parameters of the crusher, namely the closed
side and open side set. The breakage function considers breakage in terms of a series of
single particle fracture events and is dependent upon the nature of the material. It is
determined from a series of laboratory single particle breakage tests.

There is significant room for improvement when it comes to selecting and optimizing
jaw crushers. The enhancement of crusher selection and the prediction/optimization of
21
its performance are dependent upon the operating parameters of the machine and the
physical properties of the material being broken. Early attempts at establishing
relationships between these components focused on the energy consumed by a crushing
machine and the resultant size reduction. Energy-size relationships themselves require an
awareness of the mechanisms of fracture, the fundamentals of which are covered in the
following section.


2.2 PHYSICS OF PARTICLE FRACTURE

The breakage of particles in any comminution or crushing machine is a difficult process
to understand. Comprehending the sub-processes that occur prior to material failure can
provide a basis for evaluating the limitations of crushing machines or set the direction for
potential improvements (Schoenert, 1972). Knowledge of the mechanisms of fracture
provides information about the final stages of strain within a particle, namely the
formation and propagation of cracks (Tkacova, 1989). This provides a basis for how to
properly characterize materials, i.e., which mechanical properties are representative of the
fracture process.

The mechanisms of fracture also affect the distribution of resultant particle sizes.
Different methods of breakage produce different product sizes based on the amount of
energy input and process of loading. A perception of the expected distribution of
resultant particle sizes can be ascertained simply based on fracture physics.

2.2.1 PARTICLE BEHAVIOR

Rocks are brittle materials and the amount of strain within a particle under load is
proportional to the applied stress. Recalling that jaw crushers induce tensile stresses
within particles, it is the tensile strength of rock materials that must be exceeded in order
for the material to fracture. Theoretically the tensile strength of an ideal brittle material
should be approximately:

10
t
E
= [2.14]

where, E is Youngs modulus.
In reality the actual tensile strength of a rock is well below the estimate given by
Equation 2.12, usually at least 100 times lower. This decrease in strength is due to the
presence of pre-existing flaws or cracks within the rock.

Pre-existing flaws and cracks within rocks act as stress concentrators. Inglis (1913)
provided a solution for the case of an elliptical hole in a stressed plate (Fig 2.11). He
found that the stress concentration near the tip of the ellipse was proportional to the size
of the ellipse and the ellipse radius. Inglis stress concentration factor can be written as:
22


2b
2a
Figure 2.11 Elliptical hole in an infinite plate
1 2
a
k

| |
= +

\ .
|
|
[2.15]
where, a is the half-length of the ellipse
is the tip radius of the ellipse.
Griffith (1921) extended Inglis work by hypothesizing that pre-existing cracks in brittle
materials would act like Inglis ellipse causing stresses to concentrate at the crack tips.
Griffith used Inglis stress analysis to establish a relationship between fracture stress and
crack size using an energy balance approach. Griffith established that the fracture of
brittle materials is due to the existence of pre-existing flaws and is an energy-controlled
process.

2.2.2 GRIFFITHS THEORY

Griffiths proposed that the failure of a brittle solid is caused by the extension of inherent
cracks resulting in a new crack surface that absorbs energy supplied by the work done by
an external force or by the release of stored strain energy from within the solid (failure
occurs when the energy supplied by an external force or by the release of stored strain
energy is greater than the energy of the new crack surface). This implies that the more
energy a material absorbs, the more resistant it is to crack extension.

Two important criterions must be met in order for brittle failure to occur according to
Grifffiths theory (I. Chem. E. Report, 1974).
1. There must be a mechanism for which crack propagation can occur. At some
point in the material, the local stress must be high enough to overcome the
23
molecular cohesive strength of the material. This is accomplished through
stress concentration around Griffith cracks or inherent flaws.
2. The process must be energetically feasible. Enough potential energy must be
released in order to overcome the materials resistance to crack propagation.
This can be accomplished by increasing the work done by external forces
acting on the material.

As noted previously Griffith used Inglis stress analysis in his energy balance
approach. Thus the total energy for infinite plate (Fig 2.11) with an elliptical crack can
be written as:

t c
U U U W U
s
= + + [2.16]
where, U
t
is the initial elastic strain energy of an uncracked plate
U
c
is the elastic energy release caused by the introduction of a crack
W is the work done by external forces
U
s
is the change in the elastic surface energy due to the new crack surfaces.
Using elastic theory expressions, the four energy components can be determined, yielding
the following equation (Whittaker et al., 1992):

2 2 2
4
2 ' ' 2
s
A a A
U
E E

a = + [2.17]
where, A is the infinite area of the plate
E is the effective Youngs modulus (E for plane stress; E/(1-v
2
) for plane strain)
is the strain

s
is the specific surface energy.
Griffith defined the fourth energy component, where
s
is a constant material property
representing the energy required to create a unit area of new crack surface.

According to Griffith the crack will propagate when an increase in its length does not
change the net energy of the plate, when dU/da equals zero. Differentiating Equation
2.14 with respect to a results, and setting the result equal to zero, gives:

2
2
4
'
s
a
E

0 = [2.18]
Equation 2.18 can be arranged to determine the fracture stress required to cause crack
initiation and more importantly the strain energy release rate, G:

2
'
a
G
E

= [2.19]
G characterizes the energy per unit area required to extend the crack and is expected to be
a fundamental physical property controlling the crack (Dowling, 1999).

Modifications of Griffiths theory have since followed, mainly in order to compensate
for plastic deformation near the crack tip. In fact, due mainly to plastic deformation, the
energy required for crack extension has been found to be 10 times higher than that
predicted by Griffith. Plastic deformation acts to relax strain energy since energy is
24
consumed in deforming the material ahead of the crack tip. The high stress that would
ideally exist near the crack tip is effectively spread over a larger region or redistributed,
resulting in a lower stress near the crack tip that may be resisted by the material. Orowan
(1949) and Irwin (1948) took this into account and included the work of plastic
deformations in calculations of the energy balance of fracture.

Despite the definition of a characteristic material property that describes crack
propagation, the importance of Griffiths work, in particular reference to particle
breakage, really lies in the connection made between the stress required for fracture and
the presence and size of cracks or flaws. Any criterion used to define or distinguish the
crushability of rock material needs to account for the presence of flaws and their
influence on the work required to initiate fracture.

Influence of particle size
Griffiths theory indicates that fracture is dependent upon the presence of inherent flaws.
Rocks, due to their geologic nature and inhomogeneity, contain a large number of flaws
and cracks, both on the macro and microscopic level. Thus in addition to the mere
presence of flaws controlling fracture, the distribution of flaws also affects the fracture of
particles. In order to describe, and even compensate for, the distribution of flaws within
brittle materials statistical approaches are generally employed.

The most well known statistical description of flaw distribution and its effect on
particle fracture is Weibulls (1939) model. Weibulls weakest link theory assumes that
the fracture of a particle is dependent only on the local strength of its weakest flaw from
which the most severe crack will propagate and is independent of all other flaws.
Weibulls model shows that the presence of this weakest flaw becomes less probable with
decreasing particle size. This is in agreement with experimental results that indicate
strength increases with decreasing particle size. As particle size decreases critical flaws
are essentially used up in order of their significance.

Particle size effect can also be explained using energy considerations. Recalling that
failure of a brittle solid occurs when the energy supplied by an external force or by the
release of stored strain energy is greater than the energy of the new crack surface, smaller
particles, which have less capacity for storing elastic energy (U
t
is proportional to
volume), will require more work by external forces, or higher stresses, in order to
fracture. Smaller particles also exhibit a more plastic response than larger ones. The
high stress levels in small particles result in irreversible deformations that modify the
stress distribution within the particle and result in coarser fragments upon fracture
(Schoenert, 1972).

The effects of size on particle fracture call in to question the applicability of strength
parameters to crushing resistance. A jaw crusher sees a wide range of particle sizes. It
also must re-crush smaller particles produced from the crushing of larger ones. The
smaller particles will require higher stresses as the flaw distribution changes with a
25
decrease in particle size. Since tensile strength mostly controls fracture initiation of
particles in a jaw crusher, characterization of a rock material using a laboratory based
tensile strength test would require either a large number of samples of varying size or
statistical rock mechanics in order to analyze size effects. A more appropriate method of
characterizing rock materials would be one that compensates for size effects. For
example rearranging Equation 2.18 into:
2 '
s
a E = [2.20]
indicates that, according to Griffiths criterion, fracture will occur when a reaches a
constant critical value determined by the characteristic material properties E, , and
s
.
Higher stresses are normalized by the smaller flaw size, a, around which they are
concentrating. When considering fracture physics as it pertains to crushers, resistance to
fracture should account for the stresses required for breakage, the presence of flaws, and
be independent of particle size.

2.2.3 FRACTURED SIZE DISTRIBUTION

The fracture process due to the point contact loading that occurs between the plates of a
jaw crusher and a particle is illustrated in Figure 2.12. The induced tensile stress along
the axis of the particle results in fracture by cleavage. The areas directly below the
loading contacts fail in compression producing abrasion fracture. Abrasion can be
thought of as type of shatter fracture (shatter is the third type of fracture common in
impact breakers). Shatter occurs when there is excess energy input and a large number of
flaws are stressed almost simultaneously or before the weakest flaw is unloaded.
Abrasion is localized shatter. There is not enough energy to shatter the entire particle but
the impact of the jaw plates on the particle does produce attrition at the load point.
Figure 2.12 Fracture caused by compression crushing
(After Kelly and Spottiswood, 1982)

Coarse progeny particles are a result of cleavage fracture and finer sized progeny are
a result of abrasion/shatter. A comparative display of the distribution of progeny sizes
26
after particle failure is represented in Figure 2.13. Since two mechanisms of fracture
occur during point contact loading, resultant size distributions from jaw crushers are
expected to be bimodal. Lynch (1977) attributed the two components as being a result of
localized fracture against the crusher face (fine size distribution) and main fracture of the
particle (coarse distribution).
Figure 2.13 Size distributions occurring due to mechanisms of fracture
(From Bowers et al., 1991)

The distribution of particle sizes after fracture is dependent on the fracture
mechanisms occurring as a result of particle loading. The mechanisms occurring during
fracture are controlled by the intensity of energy applied to the particle. The physics of
fracture and quantification of the energy required for fracture have been explained by
Griffiths theory. Since the same mechanisms control product size, there is expected to
be a relationship between the distribution of progeny sizes and a particles resistance to
fracture (characterized by some form of Griffiths criteria).

27
2.2.4 CONCLUSIONS

A thorough understanding of fracture physics and the mechanisms that drive fracture
initiation is necessary in the evaluation and improvement of jaw crushing operations. It
is also important to be aware of fracture mechanisms when using material properties or
indexes to describe the particle breakage process or determine crusher performance.

Brittle materials fail at stress levels well below what is predicted based on stress-
strain behavior (Youngs modulus). The presence of cracks accounts for the markedly
lower strength. Rocks are brittle materials and by their nature also contain inherent flaws
(i.e., grain boundaries, voids/pores, etc.) on both small and large scales. These cracks act
as stress concentrators, with the increase in stress at these locations being proportional to
the length of the crack and the crack tip radius.

Griffith used Inglis stress analysis of an elliptical crack in an infinite plate to
determine a relationship between crack size and fracture stress. Griffith hypothesized
that fracture occurs when the energy supplied by an external force, or by the release of
stored strain energy, is greater than the energy of the new crack surface. Using an energy
balance approach Griffith was able to define the strain energy release rate G, a material
property that characterizes the energy per unit area required to a crack. Griffiths work
also defined a constant critical value of a that when met will result in fracture. The
importance in terms of crushing operations is that fracture is dependent on applied loads
and the presence and size of flaws, and both need to be accounted for when
characterizing a rock materials resistance to fracture.

It has been shown that, for a given material, as particle size decreases strength
increases. This is due to the distribution of flaws within the material. Weibulls weakest
link theory states that the strength of a particle is dependent upon its most critical flaw.
Fracture initiates from that flaw independently of all other flaws within the particle. As
the size of the particle becomes smaller the existence of such a critical flaw becomes less
probable, thus the increase in strength with decreasing particle size. Defining a
materials resistance to fracture should account for the effect of size or be independent of
it.

Since the mechanisms of fracture also control the distribution of progeny particle
sizes and specific fracture mechanisms produce specific fragment sizes, particle physics
is the foundation for the first models/equations used to describe size reduction processes.
Energy-size reduction relationships are related to Griffiths energy criterion and the
presence of Griffith cracks. The energy criterion states that enough potential energy must
be released in order to overcome a materials resistance to crack propagation, requiring
an increase in the work done by external forces acting on the material. This is the amount
of energy input into reducing the size of a particle. The amount of size reduction, or the
size distribution resulting from fracture, is dependent upon the presence and distribution
28
of Griffith cracks. Energy-size reduction relationships are a natural progression of the
physics of particle fracture and form the principles of comminution.


2.3 FRACTURE ENERGY AND SIZE REDUCTION

Relationships between energy and size reduction are concerned with the amount of
energy input by a comminution machine and the degree of size reduction attained. It was
pointed out in section 2.2.2 that different sized particles require different amounts of
energy in order to be fractured. The effect of particle size on energy input was used to
develop the first relationships between energy and size reduction.

It is now generally accepted that established relationships between energy and size
reduction can be expressed using the following singular equation (Charles, 1957):

d
d
n
d
E K
d
= [2.21]
where, E is the specific energy necessary to supply the new surface energy
K is a constant
d is the particle size
n is a value describing different size ranges.
Equation 2.21 is the general form energy-size relationships proposed by various
researchers. The principles behind their suggestions and their specific equations follow.

2.3.1 LAWS OF COMMINUTION

The work of Von Rittinger, Kick, and Bond constitute what are commonly referred to as
the laws of comminution. Von Rittinger (1867) stated that the breakage energy is
proportional to the area of the new surfaces produced and that the energy requirement
remains constant for a unit of surface energy produced. Von Rittingers law is:
( )
2 1
E K A A = [2.22]
where, A
1
is the specific surface area of the initial particle
A
2
is the specific surface area of the final particle.
Since the specific area is inversely proportional to the diameter, Equation 2.22 can be
rewritten as:

2 1
1 1
E K
d d
| |
=

\ .
|
[2.23]
where, d
1
is the initial particle diameter (mean size for a distribution of particles)
d
2
is the final particle diameter (mean size for a distribution of particles).
Von Rittingers theory is similar to Griffiths since it relates the energy input into
creating new surfaces (cracks) to the surface area of the cracks. However, it only
accounts for the energy required to pull apart molecular bonds, and as Griffith showed
there are many other energy aspects to overcome.
29

Kick (1883) postulated that the energy required for breakage is proportional to the
size or volume of the particle and that the energy requirement remains constant for
equivalent geometrical changes. Kicks equation is:

1
2
ln
d
E K
d
= [2.24]
The problem with Kicks law is that it assumes the energy required to achieve a certain
degree of size reduction will remain constant for equivalent changes in particle volume.
The effect of size on the amount of energy required for fracture, as discussed in section
2.2.2, is not considered by Kick. But smaller particles will require more energy to
fracture and the energy requirement will increase as particle volume decreases.

Bond stated that the energy input is proportional to the new crack tip length produced
and is equal to the work represented by the final particle size minus the work represented
by the initial particle size. His law of comminution is in fact an empirically derived
relationship based on a series of grinding tests. A specific form of Bonds equation is
given on page 19 in Equation 2.8. A general form in relation to Von Rittingers and
Kicks is:

2 1
1 1
E K
d d
| |
=

\ .
|
|
a.
b.
c.
[2.25]
In some respects Bonds law resolves Kick and Von Rittingers theories. Bond argued
that crushing is concerned with both surface area (Von Rittinger) and particle volume
(Kick). The volume of the particle is proportional to the amount of stress absorbed and
stresses concentrate on the surface, which eventually propagate into cracks. Bonds work
attempted to follow Griffiths theory since the premise is that when local deformation
exceeds the critical strain a crack tip forms, the surrounding stress energy flows to it, and
fracture follows (Pryor, 1974). However, Austin and Brame (1983) found that Bonds
work has no factually based relation to Griffiths theory. Bond did introduce the effect of
material properties though, noting that different materials will have different resistances
to a reduction in their size (Equation 2.6 included the work index, WI, of the material
being crushed).

The three laws of comminution have been shown to be applicable over different size
ranges (Figure 2.14). For particle sizes encountered during crushing Kicks law is most
applicable. Von Rittingers law fits fine size reduction fairly well and Bonds empirical
law of course corresponds to the grinding range. Despite the inability of each law to
fully characterize an energy-size reduction relationship for the entire spectrum of size
ranges, they do recognize some important aspects (Manca et al., 1983):
The relative importance of the energy expended
The importance of reduction ratio
The importance of the feed and product size distribution
30
d.
e.
Figure 2.14 Size ranges applicable to Von Rittenger, Kick, and Bond equations
(After Hukki, 1961)
The effects of different conditions of stress application for different size
reduction devices
The influence of the characteristics of the material.

Advancement of the three laws has included relating energy requirements to non-
uniform particle size distributions as well accounting for changes in particle strength.
Holmes (1957) modified Kick law in order to account for elastic deformation and the
effect of reduced particle size on strength. For a given reduction ratio and product size,
Holmes considers the exponent n in Equation 2.21 a measure that expresses the degree
of variation in particle strength with variations in size, and is specific to the material and
the manner of stress application. Von Rittinger, Kick, and Bond assumed that the shape
of the size distribution remains fairly constant, which is not the case, particularly for
crushers (Napier-Munn et al., 1999). Charles (1957) included the size modulus from the
Schuhmann distribution function in his equation for energy. His equation relating energy
requirement to product size distribution reduction is:
31

1
1
1
1 1
n
n
K
E
n n d

|
=

+
\ .
|
|
[2.26]
where, is the Schumann size modulus
is a constant in Schumanns distribution function
d is the particle size.
Equation 2.26 describes the relationship between energy and size reduction and includes
all the parameters upon which energy input is dependent on, but it only applies to simple
comminution systems and falls short of being applicable over a wide range of sizes.
Although Charles equation itself is not universally applicable, it also offers the
theoretical proof that a general law valid for all comminution processes cannot exist and
that every rock fragmentation process requires the derivation of a specific analytical
expression (Manca et al., 1983).

2.3.2 CONCLUSIONS

A number of researchers have investigated the relationship between energy input and size
reduction in an attempt to establish an equation or model capable of predicting the energy
requirements of comminution machines (for a given degree of size reduction). The laws
of Von Rittinger, Kick, and Bond do not describe the energy-size reduction relationship
for all size ranges but each has been found to pertain to some particle size class.

The three laws are related to Griffiths theory and seem to be founded in fracture
physics. However, Von Rittingers law only considers surface area, thus its application is
to fine sizes where volume effects are minimal. Kicks law considers particle volume but
disregards the effect of flaws or cracks on particle strength, the distribution of which
tends to decrease with size, making smaller particles more difficult to fracture. Kicks
law is applicable to coarse particle sizes. Bond considered both surface area and volume
in his theory and actually argued that it was related to Griffiths theory. That argument
has been shown to have no basis, but since Bonds equation was empirically derived from
grinding tests his model fits particle sizes typically found in that application.

Bond introduced the concept of work index to compensate for the reality that
different materials exhibit different resistances to a reduction in their size. Additional
work demonstrates that energy-size relationships need to account for non-uniform size
distributions as well as variability in particle size, strength, and load application. Yet
inclusion of these factors still does not provide a singular relationship capable of
predicting energy requirements for any range of size reduction. In reality a common law
valid for all size reduction processes probably does not exist and empirically derived
models need to be fully developed for particular breakage processes.

The complexity of the energy-size reduction relationship has lead investigators to
seek an alternative method of assessing the energy requirements and product distribution
of size reduction devices. In the past 40 years a method has been developed in order to
32
investigate the science of size reduction. The method is based on the fracture physics
discussed in section 2.2 and focuses on: fracture mechanisms, crushing under controlled
conditions, packed bed crushing, measurement of energy needed for fracture, fragment
size distribution, new surface area produced, mathematical simulation of crushing
processes, and practical applications (Rumpf, 1966). The approach used to investigate
these issues is referred to as single particle breakage analysis.


2.4 SINGLE PARTICLE BREAKAGE ANALYSIS

The breakage process occurring in a crusher, or any comminution device, is a series of
single particle crushing events. A particle is loaded and fractures only under the stresses
applied to it. The product coming out of the crushing device is just a collection of
progeny particles from numerous single particle events. An understanding of single
particle fracture, which is the fundamental process of comminution, is of great benefit.
Additionally it provides a means to separate a materials fracture behavior from the
operating conditions of comminution machines. The three laws discussed previously are
limited to certain methods of comminution and include characteristics of those methods
(Krogh, 1980). Characterization of a materials fracture behavior (i.e, by determining
values like fracture energy, fragment size) should be independent of the eventual method
of comminution used.

Although the concept of applying fracture physics, through single particle breakage,
to comminution processes was pioneered by Rumpf beginning in the 1960s, the first
single particle tests where conducted in the 1930s (Carey and Bosanquet, 1933). The
purpose of these tests was to determine the energy necessary to reduce coal and gypsum
particles (ranging from 2 to 50 mm) to a certain size. But most of the work at this time
focused on the validation or refutation of Kicks and Von Rittengers theories. Using
compression devices (to calculate input energy) and gas adsorption techniques (to
determine surface area), the relationships suggested by Kick and Von Rittenger were
tested. But since Rumpfs proposal that the fundamental areas of comminution to be
studied should include fracture physics and the material characteristics related to
breakage, a great deal of progress has been made in employing results from single
particle breakage analysis to improvements in actual crushing and comminution
applications. The following outlines the concepts behind single particle breakage, the test
methods employed, and the related data used to characterize a materials response to
fracture.

2.4.1 CONCEPTS OF SINGLE PARTICLE BREAKAGE

Single particle breakage (SPB) is the study of fracture physics (section 2.2). The state of
stress arising in a particle loaded by a comminution machine is dependent on the size and
distribution of cracks or flaws within the material and the deformation behavior of the
material. The flaw size and distribution are dependent on particle size and its
33
homogeneity. And the type of breakage device affects the type, direction, and number of
contact forces acting on the particle, and the velocity at which they are applied. Figure
2.15 shows the number of events occurring during the breakage of a particle and their
interdependence.
Figure 2.15 Phenomena that effect single particle fracture
(After Schoenert, 1987)
The presence and effect of flaws on particle fracture has been discussed in detail, as
has the effect of particle size (section 2.2.2). It is important to note again though that
rock materials are inherently flawed and only at very small particle sizes will the
presence of flaws be less critical. At those fine sizes the behavior of the particle becomes
inelastic. In terms of single particle breakage, since the determination of progeny size
distribution is of great importance, the importance of flaws is two-fold. In addition to its
influence on fracture, the crack pattern arising out of flaw propagation will determine the
size and the shape of the progeny fragments and the new surfaces created (Schoenert,
1979).

One aspect of the behavior of a stressed particle that has not previously been
discussed is the effect of deformation rate. Although single particle breakage isolates the
material behavior from comminution machine operating parameters, there is some
differentiation between slow compression and impact compression. The deformation rate
in a crusher is between 0.1 and 10 m/s, as compared to an impact mill that might deliver a
rate between 20 and 200 m/s. Single particle breakage analysis applicable to jaw crusher
processes is generally considered slow compression (but only in a relative sense, a more
appropriate classification might be slow impact). In slow compression elastic waves
34
arising from load contact do not affect the breakage of the particle, nor does the
deformation rate, as long as the particle behaves mainly elastically (Schoenert, 1991).

The stress field generated in single particle crushing is dependent upon the direction
and number of forces applied. The stress field arising in a particle under load has been
studied mainly for the case of an elastic sphere since the stress-strain behavior of
irregularly shaped particles cannot be calculated. When an elastic particle is loaded by a
contact force, the area underneath the contact develops a cone-shaped zone of stress
concentration in which particle degradation is severe compared to the space outside the
zone (Prasher, 1987). Fine fragments originate mainly from these cone-shaped zones,
with coarse fragments coming from the zones away from the contact area. This
corresponds to the situation of a particle under localized compression (Figure 2.3) and the
resultant fracture pattern (Figure 2.12). When multiple forces act on a particle the result
is more fracture since cracks grow under the contact areas and propagate towards other
contacts. Since jaw crushers perform arrested crushing particle stressing is through two
contacts only, between the jaw plate and particle (see section 2.1.1).

Figure 2.12 does not take into account what occurs after initial breakage. Since the
throw (crushing stroke) of a jaw crusher is fixed, particles are often loaded beyond the
point of first fracture. It is important in single particle breakage to distinguish between
the first stage of primary fracture and the second stage in which a group of fragments are
compressed; with fragmentation continuing until the reaction stress of the fragments is
equal to the applied stress (Schoenert, 1987). The amount of fragment particles crushed
in the second stage is dependent on the material hardness (harder materials will have
fragments that leave the crushing chamber at higher velocities) and the loading velocity.

2.4.2 TEST METHODS

There are basically three ways to reduce the size of particle, by slow compression,
impact, and abrasion/shear. Since size reduction processes are carried out in various
ways there are different methods of conducting single particle breakage (SPB) analysis,
including: impact by falling media, dropping the particle on a surface, and slow
compression (Krough, 1980). Single particle breakage analyses utilizing slow
compression and impact crushing are most applicable to the conditions of jaw crushing
and are the focus here. The distinction between slow compression and impact is the rate
of load application. Slow compression tests occur over a time scale of approximately 1 to
10 seconds, while impact tests can take as little as 110
-5
seconds to complete (Peters Rit
et al., 1983).

Slow compression is simple and has the advantage of being easy to execute (Mehrim
and Khalaf, 1980). More recently, single particle tests used to determine the behavior of
rock have evolved to include pendulum and drop weight tests (Bearman et al., 1997).
The Comminution Center at the University of Utah has made variations to these tests,
developing the Ultrafast Load Cell Apparatus. The Julius Kruttschnitt Mineral Research
35
Center (JKMRC) has also specialized in the use of similar methods (Narayanan and
Whiten, 1988).

Slow compression
Slow compression tests are essentially compressive strength tests but with slightly faster
loading rates (i.e., uniaxial compressive strength test might be run at 0.001mm/s while
slow compression for single particle breakage might be 0.1 mm/s). A particle is placed
between two surfaces, an axial force is applied, and the particle is loaded until failure. A
typical set-up employs a materials testing system to measure the force and displacement
(using a load cell and linear voltage displacement transducer).

Slow compression was developed as a method of single particle breakage analysis
because it breaks particles in the most efficient manner possible. No more energy than is
required to fracture the particle is applied. Bergstrom conducted experiments in which
95% of the applied energy was determined to be strain energy being absorbed by the
particle (Bergstrom et al., 1962; Bergstrom and Sollenberger, 1962).

Although some researchers have conducted slow compression SPB tests on
cylindrical core specimens, that particle shape is not representative of what is seen in
commercial applications, and due to the large loading surface area of a cylinder the
induced stress field is not that same as seen under point contact loading. Most research
using slow compression has been conducted using spheres or irregularly shaped particles
(Hanish and Schubert, 1986; Yashima and et al., 1979; Arbiter et al., 1969). In those
cases the particle size ranged from 2.5 mm up to 30 mm.

Data collection is straightforward. The energy required for fracture is determined
from load-displacement data collected during testing. The fragmented particle
distribution is determined from sieve analysis of the tested particles. For irregular
particles it is difficult, if not impossible, to determine the particle strength. For spheres
elastic theory can be used to relate the applied load to the stress field induced in the
particle.

Slow compression tests have given way to impact tests such as the drop weight,
pendulum, and falling media because of a need to achieve higher deformation rates and
because breakage devices that supply excess energy are more analogous to commercial
applications. A great deal of research has focused on single particle breakage relative to
ball mills and grinding mills, thus the development of various impact crushing tests.
Despite the misnomer as slow compression machines, jaw crushers apply loads over a
time period of between 0.2 to 0.3 seconds (based on typical operating rpms). The
corresponding rate of deformation (dependent also on the stroke) is much faster relative
to previous slow compression SPB research, and can more readily be simulated using
impact type single particle breakage tests.

36
Ultrafast load cell
The Ultrafast Load Cell (UFLC) was developed in order to mimic the loading and
fracture of particles in a tumbling mill, which occurs within a very short period of time
and from a wide range of drop heights (Hofler, 1990). The device was developed at the
Comminution Center at the University of Utah (Weichert and Herbst, 1986).

The UFLC has a sampling rate of 2 s enabling it to pick up data from impacts that
typically occur over a period less than 1500 s. The UFLC uses the propagation of
elastic waves to measure the force acting on a particle under impact loading. A drop
weight impacts a particle resting on top of a steel rod resulting in compressive strain
wave that propagates through the rod (Figure 2.16). Strain gages on the rod are used to
transmit a signal to a digital oscilloscope. A computer is then used to calculate the force
versus time trace.
Figure 2.16 Ultrafast load cell configuration
(From Herbst and Lo, 1992)

The energy consumption is determined in the same manner as a slow compression
test, by integrating the force-displacement plot. The fragmented particles are collected
and sieved in order to determine the progeny size distribution. The particle strength can
be determined from the maximum force applied (assuming the particle is essentially a
sphere).

37
The UFLC has been employed successfully in SPB testing. Bourgeois (et al., 1992)
broke 100 particles in the 3.35 to 4 mm range in order to determine the mass specific
breakage energy, breakage function, and particle strength. But these tests are relevant to
ball mill conditions and there has been no application of the UFLC to slower velocity
impacts seen in crushers.

Hopkinson pressure bar
The Hopkinson Pressure Bar (HPB) was originally employed to simulate the energy
levels commonly encountered in cone crushers (Napier-Munn et al., 1999). The purpose
of the HPB is to enable the resolution of the force experienced by a rock as it is
dynamically loaded in compression (Briggs and Bearman, 1995). It is also used to
measure the energy required to initiate failure in a particle.

The HPB is similar to the UFLC but is aligned horizontally. Figure 2.17 shows the
test set-up. A horizontally aligned steel bar with a rock sample attached on its end is
impacted by a smaller bar. A spring mounted on the small bar is compressed to a known
distance allowing for control of the impact velocity. An optical sensor is used to measure
the speed of the impact bar at the point of impact. As with the UFLC strain gage bridges
are used to resolve the force versus time trace.
Figure 2.17 Hopkinson pressure bar experimental set-up
(From Briggs and Bearman, 1995)

Since both bars are fitted with strain gages, the lost strain energy can be subtracted
from the input energy, allowing for the calculation of actual fracture energy.
Additionally the particle strength can be determined from the time at which a sudden
drop in force is observed on the force versus time plot. Currently the HPB is used to
determine the energy required for breakage and although it has not been used specifically
for single particle breakage analysis it may aid in crusher design (Napier-Munn, 1999).

Other investigators have employed HPB type methods. Shockey (et al., 1974) used a
gas gun to develop an approach for predicting the fragment size distribution in rock
resulting from dynamic loads. Santurbano and Fairhurst (1991) employed a similar
38
device in order to develop a mechanistic explanation of fragmentation by impact. The
rock gun uses pneumatic air instead of a spring to accelerate a rock particle down a
hollow cylinder from which it discharges and impacts a Hopkinson bar. The velocity of
the particle is determined using paired photocells and reflectors. The study by
Santurbano and Fairhurst considered the energy required for fracture and observations
were made about the resultant fracture patterns but no determination of fragment size
distribution was made.

Drop weight tests
Drop weight tests are the simplest of the single particle breakage tests. Various types of
drop weight devices have been employed due to their simple configuration (Fairs, 1954;
Arbiter et al., 1969; Jomoto and Majima, 1972; Jowett and Van Der Waedern, 1982;
Pauw and Mare, 1988).

A weight is dropped from a known height onto a particle resting on a hard surface.
The kinetic energy is determined from the mass of the drop weight, the height from
which it was dropped, and gravity. The potential energy of the weight is transmitted to
the particle, and if high enough the particle fractures. By adjusting the drop height and/or
the mass of the drop weight the input energy level can be altered. The JKRMC uses
combinations of drop height and drop weight mass that represent an energy range of
0.001 to 50 kWh/t. Figure 2.18 depicts the JKRMC drop weight test set-up.
Figure 2.18 JKRMC drop weight test set-up
(From Napier-Munn, 1999)
39
Since the drop weight comes to rest on top of crushed fragments after breakage, the
actual applied energy is the initial potential energy of the drop weight minus the potential
energy it retains at the offset height. The progeny fragments are retained for
determination of cumulative size distribution.

Pendulum tests
Pendulum tests employ twin pendulums. The input pendulum is released from a certain
height and impacts a particle attached to the rebound pendulum. Figure 2.19 illustrates a
twin pendulum set-up. The specific fracture energy is the calculated by determining the
difference in the energy of the input pendulum before impact and the energies of the
rebound and input pendulum after impact (Narayanan, 1985).
Figure 2.19 Twin pendulum test set-up
Impact Pendulum
Rebound Pendulum
Rock Particle

Bonds crushability test for determining the work index is a twin pendulum test
(section 2.1.2). Gaudin and Hukki (1946) extended the use of the pendulum test to
determine the size distribution of fragmented particles. At the JKRMC two twin
pendulum devices were developed in order to determine the energy available for
breakage and the resultant size distribution for a wide range of input energy levels
(Narayanan, 1985). A twin pendulum device was also developed by Allis-Chalmers, and
found to accurately predict rock crushability and crusher product size distribution
(Moore, 1982). However, Napier-Munn (et al., 1999) has found that twin pendulum
devices are restricted in their energy and particle size range, and that operating the device
is time consuming.

2.4.3 DATA COLLECTION

The analysis of single particle breakage is concerned with characterizing the crushing
characteristics of a material and relating those characteristics to the energy required to
40
fracture the material. A materials crushing characteristics are measured using some
parameter of material strength or resistance to fracture and the fragmented size
distribution, which are in turn used to develop energy-based relationships. Schoenert
(1991) has recommended that the fracture properties of a material should be characterized
in terms of three fundamental properties: particle strength, mass specific breakage
energy, and breakage fragment size distribution. Ultimately these characteristic and their
inter-relationships can be used, in conjunction with comminution machine parameters, to
predict and optimize the performance of size reduction processes. The following
discussion focuses on some of the parameters obtained through SPB used to describe the
three fundamental fracture properties.

Product size distribution
Fracture of a single particle results in a collection of daughter fragments. A number of
SPB tests are conducted on particles of the same material in order to group the fragments
together and determine their size distribution. Figure 2.13 showed the distribution of
fragments that results from each of the three fracture mechanisms: cleavage, shatter, and
abrasion (section 2.2.3). Based on those distributions early researchers attempted to use
mathematical expressions to describe size distributions. Gilvarry (1961) derived a
Poisson distribution based on the Griffith flaws associated with fracture of a particle.
The Gilvarry equation for product size resulting from single particle fracture is:

2 3
1
x x x
k j i
y e
(
| | | | | |
( + +
| | |
( \ . \ . \ .

= [2.27]
where, k is the average spacing of activated edge flaws
j
2
is the average amount of surface containing one activated surface flaw
i
3
is the average volume corresponding to an activated volume flaw.
When surface and volume flaws are ignored the Gilvarry equation reduces to the well-
known Rosin-Rammler distribution, and for very small values of particle size, x, it
simplifies to the Schumann equation (Gilvarry and Bergstrom, 1962).

Other researchers have proposed similar mathematical expressions. Gaudin and
Meloy (1962) derived a size distribution expression for single particle fracture using the
binomial distribution. Klimpel and Austin (1965) combined the approaches of Gilvary
and Gaudin and Meloy and developed a general expression for product size distribution.
Despite the attempts to describe the size distribution of a single particle fracture event on
theoretical bounds, none of these models reproduce the size distribution and most
researchers now agree that empirically derived functions are adequate (King, 2001).

Empirically derived expressions for product size resulting from SPB are called
breakage functions. Kelsall and Reid (1965) used a breakage function that produced a
nearly straight line on a log-log scale and that had a distribution modulus near 1. Herbst
and Fuersternau (1972) generated a distribution that deviated from a straight line but did
find that fine particle sizes were characterized by the Schumann distribution and a
distribution modulus of 0.72. Based on these, and similar findings, the breakage function
41
has been shown to be dependent only on the ratio of the progeny fragment sizes to the
initial particle size:
( )
1
1
,
n
i
i
d
B d d
d
| |

|
\ .
[2.28]
where, B is the breakage function
d
i
is the particle size
d
1
is the initial particle size
n is the distribution modulus.
The concept behind the breakage function is that the progeny population is made up of a
mixture of separate populations (the populations occurring as a result of cleavage, shatter,
and abrasion) and that it is independent of initial particle size. Different values of n
account for the larger fragments produced by induced tensile stresses, the smaller
fragments produced by compressive stresses directly under the area of loading, and/or the
fragments produced by abrasion.
Figure 2.20 A typical breakage function and corresponding parameters
0.01 0.1 1
0.01
0.1
1
n
2
= slope
n
1
= slope
1-K
K

B'
d/d
1

A typical product size distribution resulting from SPB is shown in Figure 2.20. The
breakage function modeled to fit the distribution in Figure 2.20 has two components, a
fine distribution resulting from localized fracture of the particle against the loading
surface and a coarse distribution resulting from main fracture of the particle (Lynch,
42
1977). For a 2 size fraction the breakage function of Figure 2.20 can be fit with the
following expression:
( ) ( )
1 2
1
1 1
, 1
n n
i
i
d
B d d K K
d d
| | | |
= +
| |
\ . \ .
i
d
[2.29]
where K is the fraction of progeny fragments that contribute to the finer fraction. Typical
measured values of n
1
lie between 0.6 and 1.3 while values of n
2
have been shown to be
in the range of 2.5 to 5 (Arbiter et al., 1969; Kelly and Spottiswood, 1990).

If a measured breakage function becomes bimodal, which sometimes occurs with
large particles that have a tendency to chip rather than shatter, a third distribution
modulus can be used to properly model the breakage function. Additionally, if size
distributions are determined for various parent particles of the same material and the
breakage functions do not normalize (i.e., show a dependency on initial particle size),
then the fraction K must be determined as a function of the parent size.

More recently the JKRMC has developed the one-parameter family of curves method
to represent SPB breakage data (Narayanan and Whiten, 1988). The parameter is the
percent of progeny particles passing a size one-tenth of the initial original particle size, or
t
10
. t
10
is employed as a characteristic size reduction parameter and is determined
primarily by the energy absorbed during a SPB test. For crushing applications t
10
is
usually in the range of 10% to 20% and for tumbling mills t
10
ranges between 20% and
50% (Napier-Munn, et a., 1999). Figure 2.21 shows a plot of the one-parameter family of
curves for a given material. Each t
n
value (% passing 1/n of the original particle size) is
0 10 20 30 40 5
0
20
40
60
80
100
t
n


%

P
a
s
s
i
n
g
t
10
%
t
2
t
4
t
10
t
25
t
50
t
75
Figure 2.21 One-parameter family of curves
(After Narayanan, 1985)
0
43
uniquely related to t
10
. Each vertical line represents a complete size distribution and if a
value of t
10
is given the full product size distribution can be reconstructed. An alternative
to the one-parameter family of curves is to use standard truncated distribution functions
(i.e., Rosin-Rammler, logistic, log-normal) that are dependent on t
10
and one or two other
parameters (King, 2001).

Fracture energy
The calculation of fracture energy is dependent upon the method of testing but in each
case it is usually determined as a mass specific energy (i.e., kWh/t). There are few
types of specific energy that should be differentiated. The specific energy required to
initiate fracture is called the specific breakage energy. The specific breakage energy is a
material property since it represents the energy stored in the particle due to elastic strain,
and it represents the minimum amount of energy required to cause fracture.
Measurement of the specific breakage energy requires the use of sophisticated devices
like the Ultrafast Load Cell. The specific comminution energy is the energy required to
produce a certain amount of size reduction or the characteristic size reduction parameter
t
10
. The specific comminution energy is determined in drop weight or pendulum tests
where there is no way to discern the time of first fracture. Specific comminution energy
is more applicable to actual operating conditions since size reduction machines
commonly supply excess amounts of energy.

SPB testing has indicated that the breakage function is related to the specific fracture
energy and the specific comminution energy. In general more energy results in a larger
portion of daughter fragments in the fine size range. This is particularly true with
specific comminution energy since the excessive amount of energy input into breakage is
dissipated through secondary fracture. Figure 2.22 illustrates the relationship between
specific comminution energy and product size (recalling that larger values of t
10
indicate
a finer size distribution). From Figure 2.22 it can be also be seen that t
10
can be related to
specific comminution energy using the following function:

( )
10 10max
1
c
bE
t t e

= [2.30]
where, t
10max
is a material specific limiting value of t
10

E
c
is the specific comminution energy
b is a material specific parameter.
Mehrim and Khalaf (1980) fit their test results with a similar exponential relationship
between size distribution and energy input. They also found that the relationship was
linear at lower levels of input energy and deviated from linear at high levels of input
energy (similar to what is occurring in Figure 2.23).

Bergrstrom (et al., 1962; 1965) found that the specific energy is inversely
proportional to the size modulus of the product distribution and confirmed that the
relationship holds for impact crushing and slow compression, as well as for irregularly
shaped particles and spheres. Since smaller size moduli describe finer particle
distributions this inverse relationship indicates that more applied energy produces smaller
44
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
0
5
10
15
20
25
30
35
40
45
50
t
10
= 49.1(1-e
-0.87E
c
)
t
10
(%)
Specific Comminution Energy (kWh/t)
Figure 2.22 Relationship between product size and specific comminution energy
(After Napier-Munn, 1999)
fragments and also proves, once again, that smaller particles require more energy for
fracture than larger ones. Another important aspect of this relationship is that although
increased energy results in a finer product size distribution, only the proportion of
material in the fine size range is changing (i.e., K from Equation 2.29) not the distribution
modulus (n
1,2
remain unchanged). Hanisch and Schubert (1986) confirmed this using
slow compression tests, stating that the distribution moduli are essentially not influenced
by stress conditions or comminution energy, and that only the mass fractions of the
product depend strongly on the comminution energy.

Particle strength
In studies of single particle fracture particles break in tension under compressive loads.
This is in agreement with the breakage processes occurring in most comminution
machines, particularly crushers. The stress distribution determined by Oka and Majima
(1970) (Figure 2.3) for localized compressive loading is essentially the distribution
determined by Hondros (1959) for the case of a disk subject to short strip loadings. The
maximum tensile strength can easily be determined using these distributions if the
applied load is known as well as the specimen dimensions.

For disk specimens the tensile strength can be determined from the following
equation:
45

2
t
P
Dt

= [2.31]
where, P is the compressive load, at failure, applied over a small, localized area
D is the disk diameter
t is the disk thickness.
It is more common in SPB testing to test irregularly sized particles. In this case the
tensile stresses cannot be determined from elastic theory. It is assumed that the contact
area is a certain fraction of the particle cross-section and that the stress resulting in
fracture is simply the compressive force acting on that fractional area. Equation 2.29 is
then used to determine particle strength.

2
4
t
P
D

= [2.32]
where, D is the particle diameter. Hiramatsu and Oka (1966) used photoelastic methods
to show that a sphere, prism, and cube all have a similar stress states when subjected to a
pair of concentrated compressive forces. They obtained the following expression for the
tensile strength of an elastic sphere:

2
2.8
t
p
P
D

= [2.33]
In this case D
p
is the distance between the loading contacts but in practice it is assumed to
be equal to the particle diameter. Jomoto and Majima (1972) and Oka and Majima
(1970) have suggested similar equations for determining the tensile strength of irregular
particles.

Experimental results indicate that particle strength is related to both energy input and
the breakage distribution. Yashima (et al., 1979) found that the reciprocal of the size
modulus of the fragmented product size was proportional to the compressive strength of
spheres by a power of 1.7 (for slow compression testing). Jomoto and Majima (1972)
suggested, based on drop weight tests, that the square of the tensile strength is
proportional to energy input and it could be used as a useful criterion under conditions of
impact crushing. Shockey (et al., 1974) measured both dynamic tensile strength and
fracture toughness of Arkansas novaculite and found that making quantitative predictions
of rock fragmentation based on the knowledge of a few determinable rock strength
properties is feasible.

2.4.4 CONCLUSIONS

Single particle breakage analysis provides a method for studying the most fundamental
process occurring in a size reduction machine. It is a physical representation of the
fracture physics associated with the breakage of a particle under loading conditions
typically seen in comminution machines. There are a multitude of factors that affect even
the breakage of one single particle and they are inter-dependent. The loading condition is
dependent on the machine type, which controls the number and direction of contact
forces, the deformation velocity, and the stress state of the particle. The stress field
46
induced in the particle also depends on its shape, the presence and distribution of flaws,
its homogeneity, and the particles stress-strain behavior. An understanding of how to
properly represent single particle breakage at the experimental level can be acquired if
proper consideration is given to these factors.

Several methods exist for the testing of single particles. Slow compression tests were
the first tests to be used and have now given way to tests capable of applying large
amounts of energy at high impact velocities (or deformation rates). The Ultrafast Load
Cell, Hopkinson Pressure Bar, pendulum tests, and drop weight tests are the most
common. The drop weight tests are the most simplistic but do allow for a wide range of
input energies to be applied representing basically all comminution applications. The
Ultrafast Load Cell has the ability to collect data very rapidly and allows for the
differentiation between the energy required to initiate fracture and the total energy input.

The purpose of SPB analysis is to characterize a materials crushability and relate it
to the energy required to fracture the material. A materials crushability is described
using the breakage function, an empirically derived function that expresses the
distribution of progeny particles that result form particle fracture. The breakage function
is considered to be independent of initial particle size and is a mixture of separate size
populations. When its discrete values are put in matrix form it can be used to help
predict/optimize the product size emerging from an actual crushing machine. The
JKRMC has popularized the use of t
10
, the characteristic size parameter. For a given t
10

the one-parameter family of curves can be used to develop the entire breakage function of
a material. t
10
can also be used with standard truncated size distributions in order to
develop a full product size distribution.

t
10
is determined primarily by the energy absorbed during a single particle breakage
test, as are the proportions of coarse and fine material. The specific energy is inversely
proportional to the size modulus of the product distribution but has no effect on the
distribution modulus. Thus for a given material an increase in energy intensity changes
only the proportion of fine and coarse fragments in the progeny distribution. An
exponential relationship has been suggested for determining the size distribution, or t
10
,
resulting from a certain level of specific comminution energy.

Schoenert has proposed that the fracture properties of a material should be
characterized in terms of three fundamental properties: particle strength, mass specific
breakage energy, and breakage fragment size distribution. Despite the realization that
some measure of a materials resistance to fracture should be intimately related to a
particles behavior in SPB, there have been only minor attempts to include particle
strength along with other SPB data. Most of these attempts have focused on the tensile
strength since tensile stresses control breakage in comminution processes. Even though
there is evidence that the nature of a material is related to the energy required to fracture
it and the resultant breakage distribution, it is generally thought that standard rock
mechanics tests or properties do not provide useful information for the design and
47
optimizations of size reduction processes. But the role of standard tests and properties in
understanding and optimizing comminution is likely to increase in the future if a link can
be made between a materials crushability as described by SPB analysis and its
mechanical response to fracture.

The breakage function determined from SPB testing is dependent upon the energy
input and the nature of the material broken. Since the energy input of a comminution
device is related to the method and rate of load application, the breakage function
parameters are most likely dependent upon some operational characteristics (such as
closed side set and speed or frequency), as well as the nature of the material broken. A
rapid method of establishing the breakage function for a given crushing process and
material may be possible if SPB is used to determine the breakage function and the
energy input for a certain set of operational settings, and those measures are related to an
inherent property that describes a materials resistance to fracture.


2.5 ROCK FRACTURE MECHANICS AND FRACTURE TOUGHNESS

Modifications to Griffiths theory have led to the development of the field of fracture
mechanics. Fracture mechanics deals with facture initiation and crack propagation, and
provides quantitative methods for characterizing the behavior of an intact material as it
fractures due to crack growth. The extension of fracture mechanics to rock is
understandable since rock masses contain cracks and discontinuities. States of stress
around these flaws cannot be predicted using macroscopic failure criteria (i.e. Mohr-
Coulomb, ultimate strength theories). In order to deal with crack propagation,
particularly in terms of intentional fracturing as in size reduction processes, rock
fracture mechanics must be used.

Although fracture mechanics has an undeniable place in rock mechanics applications,
it was not developed for geomaterials. It should be recognized that differences exist
between fracture mechanics for man-made materials (metals) and rock fracture
mechanics, particularly in basic material response and engineering application.
Whittaker (et al., 1992) gave a comprehensive list and explanation of these differences,
which can be summarized as:
1. Stress state Many rocks structures are subjected to compressive stresses as
opposed to tensile stresses. However, in comminution and crushing the
induced stress state is tensile (from point-load compression) and thus tensile
fracture is seen in rock.
2. Rock fracture Rock materials usually fracture in a brittle or quasi-brittle
manner and usually do not exhibit plastic flow.
3. Fracture process zone (FPZ) Non-elastic behavior ahead of a crack tip in
rock takes the form of micro-cracking as opposed to excessive shear stresses
and the resultant plastic process zone seen in metals. If the size of the FPZ is
small then linear elastic fracture mechanics applies.
48
4. Crack surface Crack surfaces in rock can be non-planar with friction and
inter-locking occurring, but linear elastic fracture mechanics assumes that no
forces are transmitted across the surface of a smooth planar crack
5. Crack propagation In rocks there is a tendency for crack propagation to
wander along grain boundaries or planes of weakness. The area of newly
created surface is then larger then the assumed fracture area.
6. Rock fracture mechanics applications In rock mechanics, as in (man-made)
materials engineering, the prevention of failure by fracture growth is a
concern. But the optimizing the generation and propagation of cracks is also a
concern as in size reduction processes. Thus the application dictates how
material parameters should be determined and used.
7. Influence of scale Due to the complicated geologic nature of rock masses,
the characterization of a rock mass is high. For the prevention of crack
growth and failure, parameters measured experimentally are of secondary
importance but for rock fragmentation applications, experimentally measured
properties are of primary importance.
8. Heterogeneity Changes in local structure and strength ahead of a crack tip
affects the continuity of crack growth.
9. Presence of discontinuities Pre-existing discontinuities affect the local stress
states and crack propagation.
10. Anisotropy Rocks can be anisotropic affecting measured fracture parameters
as a function of crack orientation.
Recognition of these variations has led to more practical and developed concepts of
fracture mechanics as it applies to rock behavior, with principles of linear elastic fracture
mechanics being extended even to rocks that behave non-linearly and much of the focus
centering on the measurement of fracture toughness.

The most fundamental aspect of rock fracture mechanics is the establishment of a
relationship between rock fracture strength and the geometry of the flaws that result in
fracture. Through this relationship an intrinsic material property that describes a
materials resistance to crack propagation can be measured. This property is called
fracture toughness. The application of fracture toughness in size reduction processes is
clear. Fracture toughness represents a critical level above which crack extension and
fracture occurs. When individual rock particles are subjected to the applied forces of size
reduction, it is most likely that the intrinsic tensile property measured as the fracture
toughness will control breakage (Bearman, 1998). Since the amount of energy input into
a size reduction process and the amount of size reduction achieved (i.e., the fractured size
distribution) are related to the type of loading and the crack pattern in the material, there
should be a relationship between these parameters and fracture toughness.

2.5.1 LINEAR ELASTIC FRACTURE MECHANICS

In section 2.2.2 Equation 2.20 was presented as a size independent expression
characterizing the resistance of a material to fracture based on the stresses required for
49
breakage and the presence of flaws. It showed that fracture initiation in a brittle solid is
controlled by the product of a far-applied stress and the square root of the flaw length
which reaches a critical value determined by the characteristic material properties E, ,
and
s.
This critical value is called the critical stress intensity factor and is denoted K
c
.

Irwin (1957) used a stress intensity approach to relate the critical strain energy release
rate G
c
to the critical stress intensity factor K
c
. Rather than follow Griffiths global
approach, Irwin considered the crack tip region, which is small compared to the rest of
the body (or plate, in reference to section 2.2) but large enough with respect to atomic
dimensions such that linear elastic theory applies (Knott, 1972). Irwin determined the
work required to close up a small portion of a crack by superimposing tensile forces
along the crack surfaces and hypothesized that this work is equal to the energy released
when the crack extends. Thus the work required to close a unit length of the crack is the
strain energy release rate and, based on the stresses and displacements occurring as a
result of the tensile forces, is equal to:

( )
2 2
1 K
G
E

= [2.34]
Since crack propagation occurs when G reaches a critical value, the critical value of
stress intensity can be defined as:

( )
2
1
c
c
G E
K

[2.35]
By demonstrating the equivalence of K and G, Irwin provided the basis for the
development of Linear Elastic Fracture Mechanics (LEFM). In LEFM the crack tip
stresses, strains, and displacements can be characterized by K as long as inelastic yielding
ahead of the crack tip is small. The advantage of LEFM is that it provides a universal
approach for determining a materials resistance to fracture, as defined by K
c
. As long as
an explicit function for the stress intensity near a crack tip is known for a given crack
geometry and loading configuration, K
c
can be measured experimentally.

2.5.2 STRESS INTENSITY FACTOR

The stress intensity factor K, alluded to in the previous section, characterizes the severity
of the crack condition as affected by crack dimension, stress, and geometry (Dowling,
1999). Determining K is based on a linear-elastic approach (hence LEFM), which
assumes the material in which the crack is located is isotropic and behaves according to
Hookes Law.

Different loading configurations at a crack tip lead to different modes of crack tip
displacement. The different types of crack deformation are generalized using three basic
modes (Figure 2.23). Mode I is the opening mode due to tension, where the crack
surfaces move directly apart; Mode II is the sliding mode due to shearing, where the
crack surfaces move over one another in a direction perpendicular to the crack front;
50
Mode III is the tearing mode also due to shearing, where the crack surfaces sliding over
one another but in a direction parallel to the crack front. The three basic modes can also
occur in combination as mixed-mode loading with the superposition of the modes
sufficient to describe most general three-dimensional cases of local crack tip stress and
deformation fields (Tada et al., 2000). Mode I is the most commonly encountered mode
in engineering applications and is also the easiest to analyze, produce experimentally on
laboratory specimens, and apply (Schmidt and Rossmanith, 1983).
Figure 2.23 The three basic modes of crack surface displacement
(After Tada et al., 2000)

Using theory of elasticity, namely the stress analysis methods of Muskhelishvili
(1963) and Westergaard (1939), the crack tip stress and displacement fields (and hence
K) for each mode of loading can be determined (for a complete derivation see Pook,
2000). With Figure 2.24 representing the coordinate system measured from the leading
edge of a crack, the Mode I stress components are given according to the following
equations:

( )
3
cos 1 sin sin
2 2 2
3
cos 1 sin sin
2 2 2
3
cos sin cos
2 2 2 2
, for plane strain
0, for plane stress
0
I
x
I
y
I
xy
z x y
z
xz yz
K
r
K
r
K
r
2
2

(
=
(

(
= +
(

(
=
(

= +
=
= =
[2.36]
where K
I
is the stress intensity factor for Mode I. The displacements at the crack tip can
be found by substituting Equations of 2.36 into Hookes Law.
51
Figure 2.24 Coordinate system for a crack tip
(After Tada et al., 2000)
It can be seen from Equations 2.36 that at the crack tip (as r approaches zero) the
stresses approach infinity, as has already been indicated by Inglis solution for stresses
around an elliptical hole in a stressed plate (section 2.2.1). Since no value of stress at the
crack tip can be given, and all non-zero stresses of Equation 2.36 are proportional to K
I,

with the remaining factors varying only with r and , the stress field near the crack tip
can be determined by giving the value of K
I
, which has a formal definition of (Dowling,
1999; Pook, 2000):

, 0
lim 2
I r y
K

r

= [2.37]
It was noted earlier that K
I
is affected by the crack size, stress, and geometry. In order to
account for different geometries Equation 2.37 can be rewritten as:
2
I
K F a = [2.38]
where, F is a dimensionless constant dependent on the geometric configuration
is the stress averaged over the gross area
a is the half-crack length.
F can generally be described as a function of loading geometry and a w where w is
defined as the maximum possible crack length. When F is determined for a given
geometry the critical value of stress intensity, or fracture toughness, can be determined as
long as inelastic yielding ahead of the crack tip is small and the conditions for LEFM are
met. Equations and values of F for a wide range of crack, specimen, and loading
geometries are determined using analytical, numerical, and experimental methods and
have been compiled in various handbooks (see Tada et al., 2000; Murakami, 1987; Rooke
and Cartwright, 1976; Sih, 1973).

52
2.5.3 MODE I FRACTURE TOUGHNESS TESTING

The critical value of stress intensity factor, K
Ic
, is determined by testing a prepared
specimen that has a crack in it. There are no standardized test methods for the
measurement of Mode I rock fracture toughness. The earliest applications of rock
fracture roughness testing employed the ASTM standard method (ASTM-E399) for
metals. Although ASTM-E399 seemed to be effective for rock testing, the general
consensus for rock testing has become that an ideal method would yield a representative
fracture toughness value and yet be simple, requiring neither pre-cracking, nor crack
length and displacement measurements, nor sophisticated evaluation techniques
(Ouchterlony, 1989). In reference to these requirements ASTM-E399 is cumbersome,
and most testing methods for rock now employ core-based specimens. The International
Society of Rock Mechanics (ISRM) has suggested two methods be established as
standardized tests in order to obtain accurate, compatible, and reproducible K
Ic
values for
rocks (ISRM, 1988).

Before covering the actual test methods used to determine fracture toughness,
consideration needs to be given to some factors that influence the applicability of
measured values and the testing procedures.

Fracture process zone
The application of linear elasticity has been shown to be valid even in cases of plastic
yielding ahead of the crack tip, as long as the amount of yielding is small relative to the
geometry of the crack and other characteristic dimensions (specimen thickness, length).
When the non-elastic field is large and dominates the crack behavior before failure, then
LEFM is not applicable and other methods of fracture analysis must be employed. Based
on the size of the non-elastic field, specimen and crack size requirements have been
developed to ensure that measured values of fracture toughness are not underestimated
(as is the case when LEFM is applied to non-linear conditions).

The fracture process zone (FPZ) in rocks is the region ahead of the crack tip that
experiences non-linear behavior. The FPZ is formed by the initiation and propagation of
micro-cracks in the vicinity of the crack tip (Figure 2.25). Models used to determine the
size and shape of the FPZ are based on the models used to describe the plastic zone in
metals. Schmidt (1980) used a maximum normal stress criterion to describe the shape of
the FPZ and found that the size of the FPZ can be given as:
( )
2 2
2
1
cos 1 sin
2 2
I
t
K
r
2

| | |
=
|
\ . \ .
|
+
|
[2.39]
where,
t
is the tensile strength of the rock material.
r, are defined by the coordinate system in Figure 2.26.
An illustration of the FPZ is shown in Figure 2.26. The characteristic size of the FPZ is
defined when is equal to zero and is identical to the plastic zone in metals under plane
stress conditions. The maximum size of the FPZ is defined when is equal to 60.
53
Figure 2.25 Development of the FPZ
(After Hoagland et al., 1973 and Whittaker et al., 1992)
Based on the previous analysis the size and shape of the fracture process zone is
independent of either plane stress or plane strain conditions. Nolen-Hoeksema and
Gordon (1987) proved this experimentally using dolomite, showing that the FPZ is the
same for a crack on a free surface (plane stress) and for a crack located within the rock
(plane strain). In comparison to the plastic zone of metals, the FPZ, and fracture
toughness accordingly, is not affected by specimen thickness (but it cannot be
subjectively small). It is however dependent upon the crack length and ligament width
(uncracked length). Lim (et al., 1994) compiled a list of minimum crack lengths required
54
Figure 2.26 Size and shape of the FPZ
(After Schmidt, 1980)
for valid fracture toughness testing of different rocks, citing that minimum crack length
depends on both material type and testing technique. However it is generally accepted
that Equation 2.39 along with aspects from the plastic zone of metals can be used to
define the specimen dimensions required to give representative fracture toughness values
(Schmidt, 1980; Barton, 1983). The requirements are:

2
2
2.5
27
32
Ic
t
Ic
t
a
K
w a
K
t

| |

` |

) \ .
| |

|
\ .
[2.40]
where, w-a is the uncracked length
t is the specimen thickness.
The requirements of Equation 2.40 are conservative, and other factors have been
proposed, but in general if the requirements given are not met the measured fracture
toughness value (i.e., the apparent fracture toughness) is considered invalid, or outside
the bounds of LEFM.

Micro-cracking ahead of the crack tip, and subsequent FPZ development, is
influenced by the microstructure or grain size of the rock. Minimum ratios of the
smallest specimen dimension to the grain size have been suggested in order to make
certain representative values of K
Ic
are measured. A minimum ratio of 10:1 is
55
recommended by the ISRM for the two suggested standard methods of testing (ISRM,
1988).

Crack requirements
In order to determine fracture toughness a specimen needs to be cracked. The crack
should be representative of a natural crack, one that is sharp and free from the effects of
residual stresses, specimen boundaries, and pre-fabricated notches (Whittaker et al.,
1992). In metals, specimens are usually pre-cracked by fatigue in order to obtain a
naturally sharp crack. In rocks fatigue pre-cracking is difficult since the loads required to
produce fatigue crack growth are usually too high (they approach catastrophic failure
levels) and care is required in order to prevent further propagation of the pre-crack after it
has been initiated.

Other methods are required to pre-crack rock materials, the simplest being to notch
the rock with a thin saw-cut. Despite the ease of notching there is evidence that notched
specimens tend to underestimate fracture toughness. Sun and Ouchterlony (1986)
showed that apparent fracture toughness values measured using notched specimens are
invariably lower than those measured from pre-cracked specimens (for the same rock
type), concluding that the use of notch length in the calculation of K
Ic
ignores micro-
crack growth prior to crack extension. Fenghui (2000) has proposed a model, based on
the size of the FPZ and notch radius, that can be used to determine the fracture toughness
based on the measured fracture toughness of a notched specimen as long as the notch
radius is not greater than the average grain size of the rock.

The most widely accepted method of pre-cracking rocks employs a chevron-notch.
The chevron-notch is a V-shaped notch that allows the length of the crack front to
increase as the crack propagates. Further propagation of the crack requires an increase in
the load, which is the condition of stable propagation, and inherent monotonic pre-
cracking is produced during testing (Sun and Ouchterlony, 1986). A sharp natural crack
is automatically formed and the resistance to propagation becomes fully developed after
initial crack growth. But the chevron-notch is not particularly easy to produce in
comparison to standard or straight-through saw-cut notches, and the decision to use one
or the other is based on the level of testing required.

Testing level
The ISRM (1988) has defined two testing levels in order to let researchers decide what
combination of screening and accuracy is best for the given application of the measured
fracture toughness values. Level I testing can be performed using portable equipment
and requires only the registration of maximum load. The associated value of fracture
toughness has the nature of an index property more so than of a material property. Level
I testing is more appropriate for screening purposes or for the rapid estimation and
comparison of fracture toughness values.

56
Level II testing requires load and displacement measurements, and thus is laboratory
based and intrinsically more complicated to perform. It is recommended for the
determination of accurate, compatible, and reproducible fracture toughness values.

Loading rate
Various loading rates have been prescribed for rock fracture testing, ranging from 0.01 to
0.03 MPa m s (Barton, 1983). For the ISRM recommended tests, the loading rate is not
supposed to be greater than 0.25 MPa m s or such that failure occurs within 10 seconds
(ISRM, 1988). Although rock strength properties exhibit loading rate dependency, there
is no consensus on whether or not K
Ic
is affected by an increase in loading rate. Based on
the FPZ it is logical to expect that dynamic loading will result in underestimated values of
fracture toughness since fracture will occur before the FPZ is fully developed. But by
testing in accordance with the recommendations of the ISRM it has been shown that
loading rate has a negligible effect on measured fracture toughness (Khan and Al-Shayea,
2000).

Calculating fracture toughness
A materials resistance to crack propagation can be quantified using fracture toughness,
K
Ic
, the critical stress intensity factor. When the stress intensity factor, K
I
, in the region
of a crack tip exceeds K
Ic
, fracture initiates and propagates until the stress intensity factor
decreases below K
Ic
. From Equation 2.38, fracture toughness is related to the applied
stress and crack length, and the function F, which is a dimensionless function dependent
on the crack and specimen geometry, and the loading configuration. When working with
applied loads and planar geometries, as is this case in rock testing, the critical stress
intensity factor is determined from the critical applied load, P
Q
, and the corresponding
crack length at failure, a
c
. The measured fracture toughness value is termed the apparent
fracture toughness, K
Q
. After K
Q
has been measured it is checked against the specimen
size requirements of Equation 2.40 in order to verify its validity.

There are various methods for the determination of P
Q
and a
c
, and they are dependent
upon the test method and the test results (for a comprehensive review see Whittaker et al.,
1992). For example, with notched specimens the critical crack length at failure is equal
to the original notch length and the critical applied load is equal to the maximum load
applied. Frequently some type of load-displacement curve is used to determine P
Q
and
a
c
. These load-displacement curves are usually in the form of applied load, P, versus
either Load Point Displacement (LPD) or, if possible, P versus Crack Mouth Opening
Displacement (CMOD). Again various methods can be employed using P vs.
LPD/CMOD curves (i.e., 5% Secant Approach, Compliance Calibration). An additional
function of P vs. LPD/CMOD curves is that they can be used to observe and even
account for non-linear behavior at the crack tip, allowing for the extension of LEFM
analysis to rocks that exhibit some non-elasticity.

57
After a series of K
Q
measurements have been validated using Equation 2.40, K
Ic
can
be determined by extrapolating the K
Q
versus crack length curve until K
Q
becomes a
constant. Since K
Ic
is independent of specimen dimensions it is a limiting value of K
Q

(which has been shown to increase with crack length).

Test methods
As mentioned previously there are two test methods suggested by the International
Society for Rock Mechanics for the determination of rock fracture toughness. They are
the Chevron Edge Notched Round Bar in Bending (CB) and the Chevron Notched Short
Rod (SR). The motivation for developing the suggested methods was to provide testing
methods that consistently yield accurate and precise K
Ic
values (ISRM, 1988). Each
method is also core based, the only viable specimen alternative as rock is often available
in the form of core pieces (Ouchterlony and Sun, 1983).

Another commonly employed test method is the Semi-Circular Bend (SCB) test. The
SCB can be prepared from rock cores and is especially adaptable to small, compact
samples that require duplicate samples to test parameters that may affect K
Ic
such as
loading rate, specimen thickness, and crack length (Karfakis et al., 1986; Chong, 1980).
Despite the suggested standard methods of the ISRM and the popularity of the SCB,
various other methods, each with its own advantages, are still used to determine rock
fracture toughness. Whittaker (et al., 1992) provides an exhaustive review of these
alternative methods.

Chevron edge notched round bar in bending
Figure 2.27 shows the specimen configuration and test set-up for the CB test. The CB
test employs a bend specimen test with a chevron-notch cut perpendicular to the core axis
(Ouchterlony, 1980). The specimen rests on roller supports and a compressive load is
applied causing crack growth and transverse splitting of the specimen. For a complete
review and background of the test see Sun and Ouchterlony (1986), ISRM (1988),
Ouchterlony (1989), Ouchterlony (1989a), and Whittaker (et al., 1992).

The core based specimen, chevron-notch, stable crack growth, ability to account for
non-linearity, and multiple testing levels (Level I and II) are the main advantages of the
CB test. However, as can be seen from Figure 2.27 the specimen geometry and loading
configuration are not simple. And despite the advantage of the chevron-notch it is still a
rather difficult pre-crack to machine. Furthermore, the advantages of the CB test seem
only to be significant in terms of Level II testing. If Level I testing is sufficient then
other more straightforward techniques are available for the rapid estimation of fracture
toughness.

Chevron Notched Short Rod
Figure 2.28 shows the configuration and test set-up for the SR test. The SR test specimen
has a chevron-notch that is cut parallel to the core axis. A tensile load is applied at the
notch mouth causing crack growth and lengthwise splitting of the specimen. For a
58
Figure 2.27 The CB specimen and testing configuration
(After ISRM, 1980 and Whittaker et al., 1992)
complete background review and detailed analysis see Barker (1977), Bubsey (et al.,
1982), Sun and Ouchterlony (1986), ISRM (1988), Ouchterlony (1989), Ouchterlony
(1989a), and Whittaker (et al., 1992).

The advantages of the SR test are the same as those for the CB test. The SR method
also uses a smaller sized specimen and according to the ISRM, two SR specimens can be
obtained from one fractured CB test (since the length to diameter requirement of the CB
test is 4:1 and for the SR it is 1.45:1). This saves material and allows for the
measurement of fracture toughness using two crack orientations. Again, the
disadvantages of the SR are that the specimen, notably the chevron-notch, is difficult to
reproduce consistently and that for Level I type testing it is too complex.

Semi-Circular Bend
Figure 2.29 depicts the specimen geometry and loading configuration of the SCB test.
The SCB test was developed and proposed by Chong and Kuruppu (1984) in order to
provide a test method that was simple to fabricate and load. The pre-crack can be a saw-
cut notch that is fatigued loaded in order to produce a natural crack, a chevron-notch, or a
very thin saw-cut notch. The kinematics of the test are similar to that of the CB test since
a vertical compressive load produces three-point bending and transverse splitting of the
59
Figure 2.28 The SR specimen and test configuration
(After ISRM, 1988 and Whittaker, et al., 1992)
Figure 2.29 The SCB specimen and test configuration
(After Karfakis et al., 1986)
specimen (Akram, 1991). A detailed description of the test is given by Chong and
Kuruppu (1984), Karfakis et al., 1986, Chong (et al., 1987), and Whittaker (et al., 1982).

Testing of the SCB specimen is simple and does not require sophisticated equipment,
and a large number of samples can be tested rapidly and accurately. But the test is
60
difficult to perform on standard NX core specimens (approximately 50 mm in diameter)
and larger diameter specimens are usually required in order to obtain stable loading and
crack growth.

2.5.4 APPLICATIONS OF ROCK FRACTURE TOUGHNESS

Because of the complicated nature of rocks and rock formations, the application of
fracture mechanics to rock fragmentation has not been straightforward. But recently
(within the last 25 years) there has been an effort to employ aspects of fracture
mechanics, particularly fracture toughness, to various types of rock breakage analyses.
This work has been able to correlate fracture toughness, the intrinsic material property
expressing a materials resistance to crack propagation, with various rock fragmentation
processes such as blasting, tunnel boring and rock cutting.

The process of rock cutting involves forcing a cutting tool into rock in order to break
out fragments from the surface. Since cutting is achieved by a fracture process it is
logical to expect that differences in cuttability of various rocks is related to the variation
of fracture material properties among the rocks (Nelson and Fong, 1986). Deliac (1986)
analyzed the chip formation due to drag picks and found that for sharp, rigid picks
operating in brittle rocks the cutting force can be expressed as a function of the fracture
toughness of the rock and the cutting depth. Guo (1990) related the fracture toughness to
the penetration rate of a diamond-coring machine and found that rocks with higher
fracture toughness are harder to penetrate resulting in lower penetration rates. Ingraffea
(ey al., 1982) has proposed using the fracture toughness of different rocks encountered
during three tunnel boring projects to predict the performance of the TBMs (Tunnel
Boring Machines). Preliminary testing showed that fracture toughness, in general,
showed less variability than other material properties (uniaxial strength, tensile strength,
and point load strength) and that fracture toughness appears to be an ideal measure to
relate to TBM performance. It has been shown that the critical energy release rate, G, is
related linearly to the penetration rates of TBMs. Since G is related to fracture
toughness by Poissons ratio and the modulus of elasticity, it follows that the penetration
rate could also be predicted by fracture toughness (Clark, 1987).

The connection of fracture toughness to some common fragmentation processes
indicates that fracture toughness could also be correlated to crushing processes. Bearman
(et al., 1991) correlated various rock strength parameters to power consumption of a
laboratory cone crusher with a statistical significance of 99.9% and fracture toughness
was among those parameters. The same study also showed that fracture toughness could
be related to product size with a significance of 95% and that K
Ic
also could be correlated
with energy-size relationships derived from single particle breakage tests. The same
correlations and level of confidence should hold true for laboratory scale primary
crushing equipment (jaw and gyratory crushers) and optimistically for actual, large-scale
industrial crushers employed at processing plants.

61
2.5.5 CONCLUSIONS

Irwins modifications to Griffiths theory, and his demonstration that the strain energy
release rate is related to the stress intensity factor, led to the development of the field of
fracture mechanics. Fracture mechanics provides quantitative methods for characterizing
the behavior of an intact material as it fractures due to crack growth. Although fracture
mechanics was developed for metallic or man-made materials its extension into rock
mechanics is natural due to the presence of inherent flaws and discontinuities within
geologic materials. These flaws control the fracture of rocks, and establishing a
relationship between flaw geometry and fracture strength is the most fundamental aspect
of fracture mechanics.

Irwins work provided the basis for Linear Elastic Fracture Mechanics. In LEFM the
crack tip stresses and displacements can be characterized by the stress intensity factor K
as long as plastic, or non-linear, deformation ahead of the crack tip is small. The stress
intensity factor characterizes the magnitude of the stresses near a crack tip in a linear-
elastic, homogeneous, and isotropic material. LEFM provides a universal approach for
determining a materials resistance to fracture since crack propagation occurs when K
reaches a critical value.

There are three general modes of crack tip displacement that are used to describe
most cases of local crack tip stress and deformation fields. Mode I is the opening mode
and it is the dominant mode in rock fragmentation. Using theory of elasticity a general
form of the Mode I stress intensity factor has been determined and shown to be a function
of the applied stress, the crack length, and a dimensionless constant, which itself is a
function of loading and specimen geometry. In order to determine fracture toughness an
explicit function that describes the stress intensity near a crack tip for a given crack
geometry and loading configuration needs to be known. Once this function is defined the
fracture toughness can be determined experimentally based on the loading condition,
crack geometry, and specimen configuration.

Fracture toughness testing of rock specimens is influenced by the development of a
fracture process zone (FPZ) ahead of the crack tip. The FPZ is formed by the initiation
and propagation of micro-cracks near the crack tip. The FPZ is similar to the plastic zone
that forms ahead of a crack tip in metals, and likewise is used to define specimen
dimension requirements, ensuring that the amount of non-elastic behavior ahead of the
crack tip is small enough so that LEFM remains applicable. Fracture toughness testing of
rock can also be influenced by the microstructure/grain size of the rock, the type of crack
used, the testing level, and the loading rate.

There are currently no standardized test methods for the determination of Mode I rock
fracture toughness. The main requirements of any test method are that it be core based,
require simple specimen preparation, be easy to load, have a straightforward fracture
toughness calculation, and yield representative and reproducible values. The ISRM has
62
suggested two methods for standardization, the Chevron Edge Notched Round Bar in
Bending (CB) and the Chevron Notched Short Rod (SR). The Semi-Circular Bend (SCB)
test is also commonly employed. Each of these tests has its advantages yet none are used
exclusively. The CB and SR specimens require a chevron-notch, which is difficult to
reproduce consistently, and the SCB test is difficult to apply to standard NX size rock
cores.

Rock fracture toughness has found wide application in processes of rock
fragmentation. Researchers have been able to correlate fracture toughness with the
penetration rates of tunnel boring machines and rock cutters. In the field of comminution
fracture toughness has been related to the power consumption and product size
distribution of a cone crusher. Based on these results it is expected that fracture
toughness can be related to the power consumption, product size, and capacity of jaw
crushers. When individual rock particles are subjected to the applied forces of size
reduction the amount of energy input and the amount of size reduction achieved are
related to the type of loading and the crack pattern in the material. Fracture toughness
measures a materials resistance to fracture based on the applied load and flaw geometry
within the rock, thus the same characteristics that control the operating parameters of jaw
crushers also control an intrinsic material property. However, in order to investigate the
use of fracture toughness in crusher applications a new test method for the rapid
estimation and comparison of fracture toughness values needs to be developed.
63
CHAPTER 3. EXPERIMENTAL PROGRAM


In order for any size reduction process to be fully characterized, material properties, or
physical expressions describing the nature of the material broken, should be related to the
pattern of breakage and resultant fragment size distribution created by crushing. It has
been suggested that a materials fracture properties should be characterized in terms of
particle strength, specific breakage energy, and the breakage fragment size distribution of
the material. It is proposed that the particle strength should be replaced by another, more
descriptive measure of a materials ability to withstand fracture. Fracture toughness fully
characterizes a rocks resistance to fracture and is dependent on the fracture strength of
the rock as well as the presence of flaws within the rock. Therefore, it is proposed that
the potential of fracture toughness to function as a predictive means for rock crushability,
and to develop fracture toughness based models for the selection and optimization of
primary jaw crushers be examined. The aim of the models is to predict the power
consumption, product size, and capacities of jaw crushers. Using the technique of
single particle breakage, discussed in section 2.4, the specific comminution energy and
fragment size distributions of various aggregates will be determined. The associated
fracture toughness of each rock type will be measured using a newly developed fracture
toughness test. Finally, a lab-scale jaw crusher will be used to test the strengths of the
laboratory-based models, and, if possible, to determine if scale-up regression analysis
between the laboratory results and actual operating conditions is necessary.


3.1 ROCK SPECIMENS

Luck Stone Corporation, the 12
th
largest producer of crushed stone products in the United
States, with operations in Maryland, Virginia, and North Carolina, has provided rocks
from five of their aggregate quarries: Boscobel, Charlottesville (Shadwell), Culpeper,
Leesburg and Spotsylvania (Figure 3.1). Additional rock was also provided from a sixth
site not currently under production, identified as the Thornburg property (located near the
Spotsylvania operation). Rock from the Culpeper site classifies into two groups, by color
(red and gray), and although the properties of each are expected to be the same, they are
to be tested independently in this study.

Each of the rock types originates in the same physiographic province. Virginia
contains five physiographic provinces: the Coastal Plain, Piedmont, Blue Ridge, Valley
and Ridge, and Appalachian Plateaus (east to west in Figure 3.2). The six Luck Stone
rocks are found in the Piedmont. The Piedmont extends eastward from the Blue Ridge to
the Fall Line, where unconsolidated sediments of the Coastal Plain cover Paleozoic-age
and older igneous and metamorphic rocks. The Piedmont is characterized by deeply
weathered, poorly exposed bedrock, and a high degree of geological complexity (DMME,
2003). Crystalline rocks are sprinkled with a few areas of much younger,
unmetamorphosed, sedimentary and volcanic rocks found in rift basins, with the largest
64
being the Culpeper Basin. A more specific geologic description of each rock type to be
tested follows and has been provided by Luck Stone (Luck Stone, 2003).
Figure 3.1 Location of rock quarries that provided specimens
(From Luck Stone, 2003)

65
Figure 3.2 Physiographic provinces of Virginia (Piedmont is green, 2
nd
from right)
(From VA DMME, 2003)
Boscobel Granite (BG)
The Boscobel operation quarries Petersburg Granite, an igneous formation about 330
million years old. The rock was formed from a molten mass and has a pink, multi-
colored appearance. Quart, orthoclase, plagioclase, muscovite, and small amounts of
dark colored minerals combine to give the granite its multi-colored appearance.

Shadwell Metabasalt (SMB)
The Shadwell operation (just outside of Charlottesville) mines Catoctin Greenstone,
formed by lava flows over 500 million years ago. The greenstone originated as basalt
and contains plagioclase and pyroxene minerals. Epidote and chlorite infused the
formation during mountain building giving the green color.

Culpeper Siltstone (CGS and CRS)
The Culpeper quarry produces sedimentary siltstone, sandstone, and shale. Siltstone is
the main rock, but the formation does grade down in particle size to shale and up in size
to fine-grained sandstone. Sediments eroded from western uplands were deposited in the
low-lying Culpeper Basin, which was formed as a result of continental land masses
pulling apart. Deposition of the sediments took place 200 million years ago, with lower
sediments gradually hardening under the effects of weight and pressure. The red colored
rock (found at greater depth) results from the hematite-cementing agents that bound the
sediment grains into rock. The upper, gray colored rock occured as a result of a lack of
oxygen during deposition that prevented the cementing agent from becoming red in color.

Leesburg Diabase (LD)
The Leesburg quarry produces diabase, sometimes referred to as traprock. The diabase is
a dense, igneous rock with greenish-black to bluish-black color. Formed by hot magma
over 200 million years ago, the rock contains mainly pyroxene and plagioclase.

Spotsylvania Granite (SG)
The Spotsylvania operation quarries a complex blend of igneous and metamorphic rock
types (Po River Metamorphic Suite). The rock types at Spotsylvania are granite-gneiss
and biotite-gneiss, and are about 400 million years old. The granite-gneiss is a white to
gray color and the biotite-gneiss is dark gray to black.

Thornburg Granite (TG)
The rock formation at the Thornburg property is similar to that of the Spotsylvania
granite. Formed by a later igneous intrusion, the rock is different in its appearance than
the Spotsylvania granite. It is a reddish granite with large mineral crystals that contrasts
with the surrounding rock. Figure 3.3 depicts each of the seven rocks to be tested.

3.1.1 ROCK PROPERTIES

A series of standard laboratory tests have been conducted on the rock types in order to
determine their mechanical properties and to provide a basis for comparison among
typical rock properties and fracture toughness as they relate to the selection of jaw
crushers. The uniaxial compressive strength, indirect tensile strength, and elastic
modulus of each rock was determined pursuant to ASTM standard test method
66
Figure 3.3 Seven quarry rocks to be tested
procedures (ASTM D2938-95: Standard Test Method for Unconfined Compressive
Strength of Intact Rock Core Specimens; ASTM D3967-95: Standard Test Method for
Splitting Tensile Strength of Intact Rock Core Specimens). The testing was conducted in
the Department of Mining and Minerals Engineering Rock Mechanics Laboratory at
Virginia Tech. Table 3.1 summarizes the test results.

Table 3.1 Mechanical properties of tested rocks
Compressive Strength Tensile Strength Elastic Modulus
Rock Type
MPa MPa GPa
Shadwell Metabasalt 142.0 18.72 29.79
Boscobel Granite 59.2 7.76 16.76
Culpeper Grey Siltstone 185.3 21.94 25.93
Culpeper Red Siltstone 155.0 22.08 23.65
Leesburg Diabase 225.7 17.61 29.51
Thornburg Granite 142.6 13.36 27.03
Spotsylvania Granite 114.3 12.44 25.38


3.2 FRACTURE TOUGHNESS TESTING

Although the International Society of Rock Mechanics has suggested that the Chevron
Notched Short Rod and Chevron Notch Round Bar in Bending be adopted as
recommended fracture toughness tests (see section 2.5.3), the use of these tests for rock
characterization and indexing purposes is not widespread. The complexity of these
existing tests is an immediate problem when considering the application of fracture
67
toughness to gauge size reduction processes. Thus, another method for fracture
toughness testing of rocks is necessary, one that will give a representative toughness
value (in accordance with Level I testing) and yet be simple, core based, with easily
reproducible crack geometry, and requiring only straightforward data collection and
evaluation techniques.

It is proposed that a wedge test utilizing an edge notched disc (END) be employed in
order to acquire fracture toughness values suitable for rapid indexing and comparative
purposes. The test will enable a large number of rock types to be tested so that
relationships among breakage energy, fragment size distribution, and fracture toughness
can be investigated in order to develop a crushing index capable of predicting and
evaluating primary crushing equipment performance. The development and experimental
verification of the END wedge splitting test, using trial rocks, is covered in the following
section.

3.2.1 DEVELOPMENT OF THE EDGE NOTCHED DISC WEDGE SPLITTING TEST

Stress intensity factors for edge notched disk
Recalling section 2.5, a materials resistance to crack propagation can be quantified using
fracture toughness, K
Ic
, the critical stress intensity factor. When the stress intensity
factor, K
I
, in the region of a crack tip exceeds K
Ic
, fracture initiates and propagates until
the stress intensity factor decreases below K
Ic
. Equation 2.38 showed the relationship
between fracture toughness, the applied stress, crack length, and F, a dimensionless
constant dependent on the geometric configuration. When working with applied loads
and planar geometries, as is this case in rock testing, the stress intensity factor can be
expressed according to:

I p
P
K F
t w
= [3.1]
where, P is the applied load
t is the specimen thickness
w is the uncracked ligament length (or the maximum possible crack length)
F
p
is a new dimensionless geometry factor, defined as:
( ) , /
p p
F f geometry a w = [3.2]

For the testing and calculation of rock fracture toughness using an edge notched disk,
the stress intensity factor and F
p
must be determined. In reference to the specimen
geometry and loading configuration given in Figure 3.4, Isida et al. (1979) and Gregory
(1979) have independently characterized the stress-strain region enclosing the crack tip
using stress field analysis for the two dimensional case, assuming linear-elastic behavior
and small-scale yielding (Murakami, 1987). For an edge cracked circular plate subject to
concentrated forces acting at symmetrical points, Isida (et al., 1979) employed the
boundary collocation method and found the stress intensity factor, F to be:
68
D
a
P P
Figure 3.4 Edge notched disk specimen and loading configuration

I
Isida
K D
F
P a
= [3.3]
Isida (et al., 1979) give values of F for a/D ratios ranging from 0.1 to 0.6 (Table 3.2).

Table 3.2 Values of a/D and corresponding value of F from Isida (et al., 1979)
a/D 0.1 0.2 0.3 0.4 0.5 0.6
F

11.488 7.721 7.051 7.451 8.636 10.990

Gregorys F equation was derived using stress functions and it is a closed form
solution. It is applicable to an edge cracked circular disk subjected to pin loading at the
crack mouth. F can be written as:

2
2
I
Gregory
K
F
P D
=
a
[3.4]
Gregory gave the function that defines F so the stress intensity factor for any a/D ratio
can be calculated directly. The expression for F is:

( ) ( )
3
2 2
1
0.966528 0.355715
Gregory
a
F
D a D a
= +

1
[3.5]

A plot of F
p
(as defined in Equations 3.1 and 3.2) versus a/D was developed in order
to compare the two solutions (Figure 3.5). F
p
in terms of Isida and Gregorys F was
created by normalizing Equations 3.3 and 3.4 against Equation 3.1 resulting in the
following expressions:

P Isida Isida
a
F
D

= F [3.6]

2
2
P Gregory Gregory
D
F
a

= F [3.7]

69
Figure 3.5 shows that the two solutions are almost identical for a/D ratios varying
from 0.1 to 0.6, and either can be used to determine the stress intensity factor for the
END. For application in rock fracture toughness testing it was determined that Gregorys
solution provides the easiest method for determining K
I
since a direct expression for F is
given, and K
I
can be determined precisely for any a/D ratio.

Load application
The previous stress analysis requires the edge notched disk to be loaded at the crack
mouth by equal and opposing forces. This loading configuration can be achieved rather
simply by using a wedge. The splitting force of the wedge, P
sp
, is the horizontal
component of the force acting on the crack mouth edge and can be determined from force
equilibrium analysis (Figure 3.6 and Appendix V). Equation 3.8 shows the splitting force
in terms of the applied vertical force, P
v
, acting on the wedge, the wedge angle , and the
coefficient of interface friction .

( )
( )
( )
1 tan
2
2tan 1 cot
2 2
v
sp
P
P

=
+
[3.8]

The wedge provides a mechanical advantage, an advantage that increases as the
wedge angle decreases. For wedging of a crack in rock the amount of friction between
Figure 3.5 Comparison of independent solutions of F
p
vs. a/D for an END
0.00 0.06 0.12 0.18 0.24 0.30 0.36 0.42 0.48 0.54 0.60
5
6
7
8
9
10
11
12
13
14
15
F
p
a/D
Gregory
Isida et al
*(for D = 2)
70
Figure 3.6 Forces acting at the crack mouth
P
sp

P
P
v
Notch

the
ble 3.3 Values of and from tilt test on hardened steel
Large grained granite Fine grained granite Dolomitic limestone Sioux quartzite
wedge material and the rock can be substantial and frictional losses may occur. The
coefficient of friction needs to be quantified for each rock type tested in order to account
for frictional losses. A tilt test can be used to determine , where the angle of sliding is
and tan is equal to . The coefficients of friction between four rock types and hardened
steel are shown in Table 3.3 (rock types and wedge material used during END wedge test
development).

Ta
.8 18.5 18.9 19.7 22
0.335 0.342 0.358 0.420

Test Specimen
he test specimen can be prepared from standard rock core, resulting in the geometry
3.4. Specimens are cut using a diamond wheel saw and the sides of the
the following
ection, but guidelines for rock fracture testing have already been discussed (section
2.5
T
shown in Figure
disk ground until parallel (or to within 0.125 mm) to each other. The edge notch is cut
using a diamond wheel saw (with a thickness of 1.5 mm) and requires a fixture that holds
the disk and ensures that the notch is cut along the centerline of the disk.

Minimum specimen dimensions for the END are investigated in
s
.3). In general, values of rock fracture toughness do not show the same type of
dependency on specimen size, namely thickness, as seen in testing of metallic materials.
But the size cannot be too small and the thickness must not be smaller than the width of
the fracture process zone. The requirements given by Equation 2.40 are used for rock
fracture testing and these are checked after an apparent fracture toughness value has
71
been measured. Since the specimens will be prepared from NX rock core the diameter
will typically be around 50 mm. The proposed thickness is approximately 25.4 mm.
Various notch lengths were proposed initially in order to investigate any relationship
between fracture toughness results and specimen size.

Wedging device
The wedging device used to apply the splitting force along the crack mouth edge consists
. The wedge angle is 11. The wedge, with approximate dimensions, is

Test set-up, loading, and measurement
he END specimen is placed on the wedge (Figure 3.8). The vertical force is applied
under displacement control by an MTS 810 closed loop servo-hydraulic materials testing
of hardened steel
shown in Figure 3.7.
Figure 3.7 Wedge configuration
13 mm
76 mm
76 mm
30 mm
6 mm
T
72
Figure 3.8 Test set-up for END wedge test
(After Donovan and Karfakis, 2003)
system affixed with an 8896 N load cell. Load and load-line displacement data are
recorded digitally by a up).

in terms of stress
intensity cannot be determined until after testing. Different rates were tested, by varying
the
be recorded in order to calculate K
Ic
. However, the load-displacement data
ould be recorded in order to verify that the maximum load is the critical applied load
that
PC ack- and also with an X-Y recorder (for b
Various loading rates, usually in terms of stress intensity per second, have been
prescribed for rock fracture testing (see section 2.5.3). The loading rate
load-line displacement, in order to investigate any dependency of the results on
loading rate. The loading rate for the END wedge test is discussed in the following
section.

In accordance with Level I rock fracture toughness testing only the maximum load
needs to
sh
results in crack propagation. For the END wedge test this involves an approximation
of the crack mouth opening displacement (CMOD). There are no plans to measure the
CMOD directly but it can be calculated from the wedge angle and the load-line
displacement. The 5% secant approach for determining the critical load has been shown
to be satisfactory for rock testing, although smaller specimens may give lower K
Ic
values.
This method, along with direct observation of the load-displacement curve (i.e., looking
for non-linear behavior), was used to verify that the maximum load was the critical load
when testing trial rocks using the END wedge test.

73
Calculation of K
Ic
fracture toughness using the END wedge test is straightforward. The The calculation of
critical load, if not the maximum load, is determined from the load and the calculated
CMOD curve (Figure 3.9). The apparent fracture toughness is calculated using the
following equation, by substituting for P in Gregorys stress analysis of the END
specimen (Equations 3.4 and 3.5) with the splitting force, P
sp
, applied by the wedge
(Equation 3.8):
( )
( )
( ) ( ) ( )
3 1
2 2
1 tan
1
2
2
2
2tan 1 cot 0.966528 0.355715
2 2
v
q
P D a
K
a
D a D a

| |
| |
|
|
= +
|
|
| +
\ .
\ .
[3.9]
The apparent fracture toughness value is considered valid (i.e., equal to K
Ic
) after

.2.2 EXPERIMENTAL VERIFICATION OF THE END WEDGE SPLITTING TEST
our rock types were tested in order to test the validity of the END specimen and test
checking that the specimen size requirements are met.
800
Figure 3.9 Example vertical load vs. calculated CMOD curve
0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 0.225 0.250
0
80
160
240
320
400
480
560
640
720
Dolomitic Limestone
Sample 4
P
v
(N)
Calculated Crack Mouth Opening Dispacement (mm)
3

F
configuration: fine-grained granite, large-grained granite, dolomitic limestone, and Sioux
quartzite. These rocks were chosen because samples of the fine-grained granite,
limestone, and the quartzite each had previously been tested using a common fracture
74
toughness test, the Semi-Circular Bend test (see section 2.5.3), providing a basis for
comparison (Akram, 1991). The large-grained granite was included in order to
investigate the influence of notch length and loading rate on fracture toughness.

Fifty-four samples were prepared in accordance with the previous section. A 15 cm
diam
ing of the specimens was done using an MTS 810 closed loop servo-hydraulic
mat
eter, diamond bladed, radial arm saw was used to cut 25 mm thick disks from 50
mm diameter rock cores. The disk sides were ground flat and parallel to each other using
a diamond surfaced grinding wheel. A 1.5 mm thick diamond blade was used to cut a
straight through edge notch along the centerline of the disc. The notch lengths varied
from 10.57 mm to 28.35 mm. Cutting, grinding, and notching of the samples took less
than 3 hours, resulting in a preparation time of approximately 3 minutes per specimen.

Test
erials testing system affixed with an 8.896 kN load cell. The tests were performed
under load line displacement control. The load-line displacement rate was 0.025 mm/s
for the limestone and quartzite, either 0.003 mm/s or 0.001 mm/s for the fine-grained
granite, and varied among the previous three rates for the large-grained granite. After
testing the loading rates were determined in terms of stress intensity. The maximum
stress intensity-loading rate occurred with the limestone and was 0.0986 / MPa m s . The
minimum occurred with the large-grained granite and was 0.0011 / MPa ssible
effects of loading rate on fracture toughness are discussed later.

m s . Po
The load-displacement data was collected and converted into splitting force (P
sp
)-
calc
fter testing, all specimen dimensions were tested against Equation 2.40 to verify that
dim
racture toughness results
the END wedge splitting and those of the Semi-Circular
ulated CMOD data in order to verify that the maximum load was also the critical
load. This required the measurement of the frictional coefficient between the rock and
the hardened steel of the wedge. For all 54 samples the critical applied load resulting in
crack propagation was also the maximum load.

A
ension requirements were met (tensile testing of the same rock types was performed
in order to determine
t
). Of the 54 samples tested, 1 fine-grained granite sample and 3
limestone samples did not meet the minimum requirement for crack length (one of the 3
limestone samples also did not meet the thickness specification). These samples were
disregarded.

F
Table 3.4 shows the results of
Bend test (Akram, 1991). Statistical analysis of the results for each individual rock type
indicates that the mean values of fracture toughness from both tests are not significantly
different (at a confidence level of 99%). From the initial analysis it seems reasonable to
expect the END wedge split test to yield representative fracture toughness values,
particularly in terms of indexing and for comparative purposes.


75
Table 3.4 Fracture toughness values from the END test and SCB test
END SCB
Rock Type
# of Specimens
Ic
K MPa m # of Specimens K
Ic
MPa m
13 0.914 3 6 0.884
Large grained granite NA NA 27 0.621 0.101
Dolomitic limestone 9 1.398 0.186 6 1.33 080 1 0.
Sioux quartzite 4 1.241 0.031 6 1.244 0.071
Fine grained granite 0.09
0.022

Based on the test results, there is some dependency on grain size. The larger grained
roc
rack length effects
cture toughness values versus a/D for the two granite rocks tested.
ks had the lowest fracture toughness values, and this follows the general observation
that rocks with smaller grains tend to have higher values of fracture toughness (Whittaker
et al., 1992). Generally, for rock testing, the ratio of smallest specimen dimension to
grain size should be at least greater than 10:1. The END specimens in this case satisfied
that requirement, but there is some question as to the effect of notch thickness. The
maximum notch width should also be considerably larger than the grain size. In this case
a notch thickness of 1.5 mm was used, which may not be large enough relative to the
granite rocks.

C
Figure 3.10 shows fra
Figure 3.10 Fracture toughness vs. a/D
0.00 0.06 0.12 0.18 0.24 0.30 0.36 0.42 0.48 0.54 0.60 0.66
0
100
200
300
400
500
600
700
800
900
1000
1100
K
Ic
a/D
Fine-grained granite
Large-grained granite
76
There is some dependency of K
Ic
on crack length (D was constant at approximately 50
mm). Usually when the crack length is long enough, the fracture process zone becomes
fully developed before failure, resulting in the consumption of more energy and a higher
fracture toughness value. Figure 3.10 seems to contradict this, as K
Ic
seems to decrease
slightly with increasing crack length, but the amount of scatter in the data requires that a
wider range of crack lengths be tested before suggesting a relationship between K
Ic
and
crack length for the END test.

Effect of loading rate
that the loading rate was varied among 0.025, 0.003, and 0.001 It was previously noted
mm/s for the large-grained granite. In terms of stress intensity, the corresponding loading
rates were 0.0283, 0.0032, and 0.0011 / MPa m s . At least 8 samples were tested at each
rate so that an analysis of variance could be performed to quantitatively assess any effect
of loading rate on K
Ic
. At a confidence interval of 99% there is no significant difference
between the fracture toughness values determined at different loading rates. For the END
wedge split test it is recommended that the load line displacement be no greater than
0.025 mm/s.

Test limitations
In section 2.5.3 it was stated that fracture toughness specimens employing a notch instead
of a sharp crack tend to underestimate K
Ic
. This is the biggest drawback to using the
END specimen. However, the purpose of the END test is to provide representative
values of fracture toughness for use in indexing different rocks quickly and easily, in
regards to sample preparation and testing. From this standpoint the limiting effect of a
notch does not seem to be an issue since the results of the END tests compared so
favorably to the Semi-Circular Bend tests. For application to Level II testing it would be
worthwhile to investigate the use of a sharp crack (i.e., a chevron notch) with the END
specimen.

3.2.3 APPLICATION OF THE END WEDGE SPLITTING TEST

The Edge-Notched Disc Wedge Splitting test gives a representative toughness value (in
accordance with Level I testing) and is a simple, core based specimen, requiring only
straight-through notch and straightforward data collection for the determination of K
Ic
.
The END test can be used for indexing and comparative purposes. The test will enable a
large number of rock types and specimens to be tested so that relationships among
breakage energy, fragment size distribution, and fracture toughness can be investigated,
leading to the development of an improved method for prediction and evaluation of
crushing equipment.

For this study the testing procedure follows the one used in the development and
verification of the END test. A loading rate of 0.001 mm/s is to be used to ensure slow,
stable crack growth. T roc e tested. The average en specimens of each k type are to b
77
specimen dimensions for each rock, along with the coefficient of friction between the
wedge and rock, are given in Table 3.5.

Table 3.5 Average specimen dimensions and coefficient of friction for tested rocks
D a t
Rock Type
mm mm mm
Shadwell Metabasalt 47.422 21.946 24.696 0.384
Boscobel Granite 47.483 22.007 25.354 0.410
Culpeper Grey Siltstone 47.597 20.668 25.042 0.386
Culpeper Red Siltstone 47.589 22.289 25.166 0.372
Leesburg Diabase 47.597 21.763 25.776 0.350
Thornburg Granite 47.160 21.237 25.588 0.398
Spotsylvania Granite 47.468 21.989 25.197 0.394


3.3 SINGLE PARTICLE BREAKAGE TESTING

In any size reduction process the breakage of any individual particle occurs
simultaneously with that of many other particles. However, each individual particle
breaks only as a result of the stresses applied to it, thus isolation of the single particle
acture event represents the most elementary process in size reduction and its study is of
lication, and the energy input. The purpose of
ngle particle breakage in this study is to derive crushing energy and product size
d
fracture toughness of the those rock types.

Th vices u single e break dies we cribed in
sectio f those te ystems used in this study. Single particle
breaka ed o g the Allis-Chalmers Energy ing Test
(HEC CT is a s confi TS m s testing
system f s eously ring c force actuator
displa to accura lculate energ ed in pa crushing
(Allis . Additi he HE em allows for flexible control of the
force ring a that a cified can be followed. It is
apable of running under displacement control at 2 m/sec, using an inverted haversine
rofile permitting close simulation of the crushing cycle in an actual crusher. The HECT
er sets, speeds, and throws, a range that
cludes essentially all crusher operating conditions, consequently distinguishing itself
fr
great importance. The characteristics of crushing that need to be measured are force
applied at fracture, total fracture energy, specific fracture energy (taking into account the
mass of the particle), and the product size distribution. The product size distribution is
used to determine the breakage function, which is influenced by the material properties of
the particle, the nature of the stress app
si
istribution data for the seven rock types, data which will ultimately be compared to the
e most common de sed for particl age stu re des
n 2.4.2. None o sting s will be
ge tests will be carri
T) system. The HE
ut usin
system
High
gured M
Crush
aterial pecially
that is capable o imultan measu rushing and
cement in order tely ca the net y utiliz rticle
-Chamlers, 1985) onally t CT syst
or displacement du test so pre-spe profile
c
p
system can simulate a wide range of crush
in
78
from other testing systems. The Department of Mining and Minerals Engineering Rock
Mechanics Laboratory at Virginia Tech is equipped with a HECT system.

3.3.1 CONCEPTS OF THE HIGH ENERGY CRUSHING TEST

The purpose of the HECT test is to obtain the breakage functions for various feed sizes
over a range of crushing energies or size reduction levels. A number of pieces of
material are broken, the crushing energy is recorded for each piece, and the products
pooled together. The size distribution of the pooled pieces is determined by sieve
analysis and the breakage function is determined in accordance with the methods outlined
in section 2.4.3.

Since the crushing energy is dependent upon the amount of size reduction and the
e
ich carries the
st material up towards a fixed platen. A load cell affixed to the upper platen measures
lly by a data acquisition system.
uck Stone provided core boxes of the rock types to be used in this study. Therefore the
e state of stress within a
article under point contact loading is essentially independent of the particle shape and,
m chanical properties of the material, the range of HECT tests is defined by various
reduction ratios. The reduction ratio is the ratio of the geometric feed size to the closed
side set of the HECT device. Allis-Chalmers recommends that HECT tests be run for
reduction ratios between 1.6 and 3.3. It is also recommended that various feed sizes,
representing what might be seen in an actual crusher, be tested.

The HECT system is very flexible and can accommodate a wide range of crusher
conditions. The HECT system requires the specification of three conditions: the closed
side set, the throw, and the speed (or frequency). These parameters are set to match the
operating conditions of the crusher application in question.

The HECT system itself measures only two parameters during testing, the crushing
force and the displacement of the hydraulic actuator. A linear variable differential
transducer (LVDT) measures the displacement of the hydraulic ram, wh
te
the crushing force. These two signals are recorded digita

The following section outlines the exact HECT test procedure, set-up, data
acquisition and analysis, and specimen geometry used in this study.

3.3.2 HECT TEST SET-UP AND PROCEDURE

Test specimens
L
specimen geometry for the HECT test is limited by the cylindrical geometry of rock
cores. Normally single particle breakage is performed using irregular shaped particles or,
in some instances, spherical particles are used. There is no evidence in the literature that
cylindrical specimens have been used previously for single particle breakage. The main
limitation of any specimen with a uniform, geometric shape is that it is not representative
of the irregular particles fed to an actual crusher. However, th
p
79
as with irregular particles, cylindrical particles fracture due to induced tensile stresses
that act along the load-line.

Table 3.6 gives the average specimen dimensions for each rock tested using the
ECT system. A cylindrical specimen with a thickness approximately half of the
Reduction Ratio 1 Reduction Ratio 2
H
diameter was chosen. The specimens are cut using a radial armed saw outfitted with a
diamond wheel blade. Approximately 20 specimens, or at least enough to exceed 2000 g
by mass (total), are prepared for each test (i.e., varying reduction ratios) per rock type.

Table 3.6 Average HECT test specimen dimensions
D t Mass D t Mass
Rock Type
mm mm g mm mm g
Shadwell Meta lt 7 basa 47.47 25.883 126.8 47.481 26.109 128.8
Boscobel Gran 2 ite 47.49 25.668 117.0 47.464 26.041 117.5
Culpeper Grey Siltsone 47.624 26.601 128.3 47.621 25.575 124.0
Culpeper Red Siltsone 47.631 26.034 127.6 47.605 25.740 125.5
Leesburg Diabase 47.587 26.575 141.2 47.608 26.038 139.1
Thornburg Granite 47.174 26.728 125.1 47.192 26.064 119.9
Spotsylvania Granite 47.454 26.673 128.3 47.431 26.069 125.2

Test equipment and set-up
The HECT system is a uniquely designed MTS 810 materials testing system (Figure
3.11). It is a closed loop servo-hydraulic materials testing system affixed with a 250 kN
load cell. The force-generating device in the HECT system is a high velocity hydraulic
actuator. The actuator, or ram, is capable of speeds up to 2 m/s and has a maximum
roke length of 152 mm. The actuator is rated at 150 kN. The displacement of the ram
s,
ith the closed side set for reduction ratio 1 being 31.75 mm and for reduction ratio 2,
16.00 mm. Base erage reduction
ratios are 1.50 and 2.97. The t istance travele is fixed
at 50.8 rdance with Allis-Chalmers ion tha l tests b un at t ow
(Allis-Chalmers, 1985). The speed of the test is to be representative of an actual jaw
cr an a spe enc a sin gle ushe the
te will be lose spe he sys uire the
sp req thu st is at ing cy Hz
(2

st
is measured by an LVDT mounted inside the hydraulic piston.

The closed side set, throw, and speed are the only parameters that need to be defined
for the HECT. The closed side set is determined based on the required amount of size
reduction. For this study it is proposed that tests be run at two different reduction ratio
w
d on the average dimensions given in Table 3.6, the av
hrow (the linear d d by the actuator)
mm in acco direct t al e r hat thr
usher. 225 rpm is verage ed/frequ y for gle-tog jaw cr r and
sts outlined here run c to that ed. T HECT tem req s that
eed be entered as a f
28 rpm).
uency, s the te set-up a crush frequen of 3.8
80
The test is run under displacement control with the actuator displacement following
an inverted haversine profile, shown in Figure 3.12. The combination of the given throw
and frequency results in an average loading rate of 386 mm/s. Breakage of one particle
takes 0.26 seconds with full-scale displacement occurring at 0.13 seconds.

Each particle is placed inside the crushing zone, as defined by the small platen affixed
to the load cell (Figure 3.13). The crushing chamber is surrounded by a plastic enclosure
that is closed and locked after a particle is loaded. The particle is then broken. After
breakage the particle fragments are cleaned from the chamber and transferred to a plastic
bag.

Data acquisition
A digital acquisition system accompanies the HECT system (upper left hand corner in
Figure 3.11) and records and analyzes the raw data sent to it during each test. The raw
data is the displacement as measured by the LVDT and the load as measured by the load
cell. The DATA 6000 Waveform Analyzer acquires this data during testing and is also
capable of processing it after testing. The DATA 6000 has a sampling rate of 10 s.


Figure 3.11 HECT system
81
82
The DATA 6000 can be programmed manually or by using a program supplied
by Allis-Chalmers. In each case the DATA 6000 is configured to collect the raw data
Figure 3.12 Displacement profile of HECT actuator
Frequency = 3.8 Hz
50.8
Time (s)
-50.8
0
0.132
A
c
t
u
a
t
o
r

d
i
s
p
l
a
c
e
m
e
n
t

(
m
m
)
Crushing Zone
Actuator
Fixed platen and Load cell
CSS + Throw
Figure 3.13 Schematic and picture of HECT loading configuration
83
fter a certain triggering level has been met and to process the data according to the
input program. For this study the DATA 6000 is programmed to calculate the crushing
energy from the force and displacement data. The DATA 6000 differentiates the
stroke/displacement signal in order to obtain the actuator velocity curve. The force data
is multiplied by the velocity curve in order to determine the power curve. The DATA
6000 then determines the points at which crushing began and ended. This interval is
bounded by the first non-zero force value and the maximum displacement of the actuator
(i.e., the point at which the actuator begins to retract). The waveform analyzer then
integrates the power curve over this interval in order to determine the total crushing
energy. Figure 3.14 illustrates a typical set of DATA 6000 recorded and processed
curves.


After each breakage event an HP 7470A plotter produces a graph similar to Figure
3.14. In addition to the plot, the DATA 6000 also provides numerical results for the total
crushing energy and the peak crushing force. These values are recorded for each
specimen. The crushing energy is given in points, where 1 point is equal to one watt-
second. The crushing energy is converted to specific comminution energy with units of
kW-hours per metric ton. Using Equation 2.31 and the peak crushing force, the dynamic
tensile strength of the specimen is calculated.

Upon completion of one series of testing (all specimens tested for a given closed size
set/reduction ration), a sieve analysis is performed on the pooled breakage fragments.
For this study a
a
2 series will be used with an initial sieve opening of 38.1 mm. The
Figure 3.14 Recorded and processed curves from DATA 6000
End of crushing
Start of crushing
83
material will be sieved down to 0.074 mm (200 mesh). A Ro-Tap sieve shaker is used to
provide uniform circular tapping motion to the sieves. The particle-size distribution is
recorded as the weight percentage retained on each of the sieves of decreasing size and
the percentage passed of the finest size. This data is converted to the weight percentage
passing each screen in order to determine the breakage function (see section 2.4.3). For
the cylindrical disk specimens of this study the parent size is taken to be the diameter of
the disk.


3.4 DATA REDUCTION AND MODEL DEVELOPMENT

The results of the Edge Notched Disk test and the High Energy Crushing Test will be
compared in order to investigate and develop relationships between fracture toughness
and specific comminution energy and fracture toughness and breakage distribution.
Since the closed side set is the only HECT parameter that is variable, the specific
comminution energy, E
c
, and breakage function, B, are expected to be functions of
fracture toughness and reduction ratio. Regression analysis will be performed using
Microcal Origin 6.0.

The fracture toughness results will also be used to modify Equation 2.12, which is
used to estimate capacity of a jaw crusher and is dependent upon a parameter related to
the nature of the material crushed. It is proposed that fracture toughness can be used to
define this parameter. Lab-scale crushing tests will be conducted in order to determine
how fracture toughness can be used in this regard.


3.5 LABORATORY SCALE CRUSHING TESTS

The power of the models developed from analysis of the END test and HECT test results
will be assessed by comparing the predicted and actual results generated from lab-scale
crushing tests. The laboratory crusher results will also be used to develop a fracture
toughness based model for the prediction of jaw crusher capacity.

3.5.1 CRUSHING EQUIPMENT

The tests will be performed using a 102 mm by 152 mm single-toggle, overhead
eccentric, Blake-type, Morris jaw crusher with a feed opening of 75 mm.

3.5.2 ROCK SAMPLES

Rock samples from Luck Stones Boscobel, Culpeper, Shadwell, and Spotsylvania
operations will be used. The samples were collected from aggregate product piles.
Figure 3.15 shows the feed size distribution of the four samples.

84

3.5.3 TEST PROCEDURE AND DATA ACQUISITION

Each test will use approximately 35 kg of material. The material is fed to the crusher and
the time required to reduce the size of the feed material is recorded. The open and closed
side sets of the crusher are noted. The crushed product is collected and a sieve analysis is
conducted in order to determine the product size distribution.
90
100
Figure 3.15 Feed size distribution of material fed to laboratory jaw
4 5 6 7 8 9 10 20 30 40
0
10
20
30
40
50
60
70
80
SMB
BG
CR
SG
%

p
a
s
s
i
n
g
Particle size (mm)


85
CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION


4.1 FRACTURE TOUGHNESS

Ten specimens of each rock type where tested in accordance with the END test procedure
set forth in section 3.2. The results are shown in Table 4.1. Figure 4.1 displays the
tatistical scatter of test data for each rock (data points represent valid tests). Appendix I
ontains complete test data.
Table 4.1 Fracture toughness results
s
c

K
Ic
( SD)
Rock Type
No. of specimens
tested
No. of valid
specimens
MPa m
Shadwell Metabasalt (SMB) 10 7 1.411 0.215
Boscobel Granite (BG) 10 5 0.601 0.083
Culpeper Gray Siltstone (CRS) 10 10 1.454 0.195
Culpeper Red Siltstone (CGS) 10 10 1.501 0.262
Leesburg Diabase (LD) 10 10 1.254 0.101
Thornburg Granite (TG) 10 10 1.045 0.037
Spotsylvania Granite (SG) 10 8 0.843 0.165

Of the seventy rock specimens tested, ten did not qualify as valid specimens. Three
specimens of the Shadwell metabasalt and two specimens of the Spotsylvania granite did
ot meet the specimen size requirements set forth by Equation 2.40. The five invalid
ecimens of Boscobel granite did not fail at the notch (i.e., crack propagation ahead of
o the friable nature of the Boscobel granite,
hich consists of large grains with multiple planes of weakness. BG samples were
hness values show a
ependence on grain size. The results in Table 4.1 support this trend. The Culpeper
d sedimentary rock and has the highest fracture toughness
alue. The metabasalt and diabase have slightly larger grain sizes and correspondingly
ure toughness.
here may be a propensity for the K
Ic
of coarse-grained rocks to be underestimated in
ained rocks when a notched specimen is used. This point will be
onsidered when comparing fracture toughness and specific comminution energy.
hat the two rocks are
irtually identical. Statistical analysis of the separate siltstone results shows that at a
n
sp
the crack tip did not occur). This was due t
w
difficult to prepare as often times the rock core would crumble or break apart while being
sawed.

As was noted in section 2.5 and section 3.2, rock fracture toug
d
siltstone is a very fine-graine
v
have slightly lower K
Ic
values. The granites follow in the same manner, with the
Boscobel granite, consisting of fairly large grains, having the lowest fract
T
comparison to fine-gr
c

Figure 4.1 indicates that the Culpeper siltstone rocks (red and gray) had very similar
results. The mechanical properties given in Table 3.1 also show t
v
86
3.0
+SD
2.5
Median value
-SD
Figure 4.1 Fracture toughness data for all valid specimens
SMB TG SG LD BG CGS CRS
0.0
0.5
1.0
1.5
2.0
K
I
c

(
M
P
a

m
0
.
5
)
Rock Types
confidence level of 99% the average fracture toughness values are not significantly
different. Unless significant differences in red and gray siltstone behavior are observed
during single particle breakage, the test values of the two rocks will be pooled and the
rocks considered as one type/sample.

For all valid test specimens fracture initiated at the notch tip. Fracture propagation
did not a surface
planar. The crack surfaces of the Spotsylvania and Thornburg granites wandered towards
lanes of foliation but never directly along them. Crack propagation occurred along
lanes of weakness represented by narrow veins of mineral infilling in the Boscobel
granite. The granites were more susceptible to changes in heterogeneity and the
continuity of crack growth was affected by changes in local structure ahead of a crack tip.
The metabasalt and diabase demonstrated somewhat more parallel crack propagation,
with the Culpeper rocks almost always exhibiting parallel propagation. All the rocks had
somewhat non-planar crack surfaces, most likely due to friction and inter-locking.

Figures 4.2 through 4.4 show the correlation between fracture toughness and the
major mechanical properties of each rock (compressive strength, tensile strength, and
elastic modulus). As would be expected, fracture toughness and tensile strength exhibit a
strong correlation since tensile stresses are responsible for the opening of flaws or cracks
lways occur in a direction parallel to the notch tip nor was the crack
p
p
87
Figure 4.2 Correlation between fracture toughness and tensile strength
5.0 7.5 .0 12.5 15.0 20.0 22.5
0.8
K
I
c

(
M
T ile strength (MPa)
2.0
1.8
1.6
1.0
1.2
1.4
P
a

m
0
.
5
)
0.0 2.5
0.0
10 17.5 25.0
0.2
0.4
0.6
ens
2.0
1.0
1.2
1.4
1.6
1.8
M
P
a

m
0
.
5
)
0.8
K
I
c

(
within rock materials, particularly under conditions of point contact loading. There does
not seem to be much of a link between K
Ic
and elastic modulus but it is interesting to note
that the Culpeper siltstones, with the highest fracture toughness values, are among the
softer rocks or lower modulus rocks. Two vertical lines have been added to Figure 4.4,
Figure 4.3 Correlation between fracture toughness and elastic modulus
0 4 8 12 16 20 24 28 32 36 40
0.0
0.2
0.4
0.6
Elastic modulus (GPa)
88
89
one at 69 MPa and the other at 138 MPa. The lines represent the cut-offs for material
ranked soft and material ranked average, for crushing applications, according to Table
2.2. Above 138 MPa rocks are considered hard and supposed to require full power draw.
In Figure 4.4 it can be seen that five of the rocks would be considered hard but that the
three rocks that showed the highest resistance to fracture are not the hardest according to
the rankings.

4.2 SINGLE PARTICLE BREAKAGE

High Energy Crushing Tests were conducted in accordance with the procedure set forth
in section 3.3. Approximately twenty specimens of each rock were tested at each
reduction ratio. The reduction ratio for the first series of tests was 1.50, corresponding to
a closed side set of 31.75 mm. The second series of tests was conducted at a reduction
ratio of 2.97 with the closed side set equal to 16.00 mm. Data collection included particle
strength (tensile), specific comminution energy, and breakage size distribution. A
comprehensive catalog of HECT data can be found in Appendix II and Appendix III.

Again, the red and gray Culpeper siltstones did not exhibit different behavior, with
the data for each rock not statistically different. From this point forward the individual
results for the red and gray siltstones will be shown since they were considered separate
rocks when tested, but for comparative purposes (section 4.3) their results will be pooled
and presented as one rock, Culpeper siltstone.

Figure 4.4 Correlation between fracture toughness and compressive strength
0 25 50 75 100 125 150 175 200 225 250
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
K
I
c

(
M
P
a

m
0
.
5
)
Compressive strength (MPa)
4.2.1 PARTICLE STRENGTH

Table 4.2 summarizes the dynamic tensile strength results from both HECT series.
Particle strength data was collected in order to compare traditional rock properties
collected under conditions of slow displacement with a strength property measured under
the dynamic or high strain rate conditions seen in a jaw crusher. The compressive
strength, tensile strength, and elastic modulus of each rock were tested under
displacement control at a rate of 0.003 mm/s. As was mentioned in section 3.3, the
crushing tests were conducted at a rate of 386 mm/s. Additionally the dynamic tensile
strength was compared to the fracture toughness in order to verify that mode I crack
opening is applicable to the crack displacement behavior of particles under point contact
loading.

Table 4.2 Tensile strength measured in HECT tests
Reduction Ratio 1 Reduction Ratio 2
Tensile Strength ( SD) Tensile Strength ( SD)
Rock Type
MPa MPa
Shadwell Metabasalt (SMB) 21.69 4.10 24.80 6.89
Boscobel Granite (BG) 11.90 5.84 10.03 3.46
Culpeper Gray Siltstone (CRS) 27.51 3.22 26.29 2.85
Culpeper Red Siltstone (CGS) 28.89 3.69 26.41 4.93
Leesburg Diabase (LD) 26.45 3.93 25.72 3.28
Thornburg Granite (TG) 17.60 4.72 16.16 2.88
Spotsylvania Granite (SG) 13.58 4.13 15.71 3.86

Care had to be taken when determining the load that caused initial fracture. The
HECT/DATA 6000 set-up only records the peak load, which in some instances occurred
after fracture had already initiated and represented a smaller daughter fragment being
loaded. In these instances the force curve produced by the DATA 6000 acquisition
system was used to distinguish the point at which fracture first initiated and the
corresponding load on the disk specimen at that time. In all other instances the peak load
recorded by the DATA 6000 represented the failure load.

On average the increased displacement rate resulted in about a 29% increase in tensile
strength for each rock. The data in table 4.2 indicates that, for each rock, there is little to
no change in the dynamic tensile strength at the two reduction ratios. The change in
reduction ratio would not be expected to influence the measured particle strength, as there
is no c he ratios, only a c peak
fractur uency
hange in strain rate between t hange in total strain. The
e load would be more affected by a change in either the crushing speed/freq
or the throw.

Figures 4.5 through 4.8 show the relationships between dynamic tensile strength and
tensile strength, elastic modulus, compressive strength, and fracture toughness. Figures
90
4.5, 4.6, and 4.7 indicate that, for the rocks tested here, the relationships/correlations
between dynamic tensile strength and tensile strength, elastic modulus, and compressive
strength are almost identical to those between fracture toughness and those latter
properties (compare to Figures 4.2 4.4). This is evident also by the strong correlation
between dynamic tensile strength and fracture toughness seen in Figure 4.8. The strong
correlations of dynamic tensile strength with tensile strength and fracture toughness again
signify that fracture under point contact loading is due to induced tensile stresses. It is
also clear that under rapid loading the magnitude of the induced tensile stresses increases
but the fundamental mechanical behavior of rocks is unchanged (i.e., the increase in
tensile strength is approximately proportional for all rocks).

The relationships among fracture toughness, dynamic tensile strength, and tensile
strength highlight the application of fracture toughness as a strength parameter. When
the proper crack opening displacement mode is identified for a given situation, fracture
toughness effectively characterizes a materials resistance to fracture. In the case of jaw
crushers, tensile stresses open cracks and the ability of mode I fracture toughness to
assess a rocks resistance to breakage is shown by the strong correlation between fracture
toughness and both tensile strength measurements. In other instances though, for
example cone crushing, particles may also be subjected to shear stresses and mode III
(mixed-mode/tearing) fracture toughness may be more applicable. The ability of fracture
toughness to characterize a rocks resistance to fracture is dependent upon an
understanding of the stress situation within the particle under the load action of the size
reduction machine.
Figure 4.5 Correlation between dynamic tensile strength and tensile strength
2.5 15.0 17.5 20.0 22.5 25.0
8
32
40
D
y
h

(
P
a
)
ength (MPa)
o 1 36 Reduction rati
Reduction ratio 2
28
M
12
16
20
24
n
a
m
i
c

t
e
n
s
i
l
e

s
t
r
e
n
g
t
0.0 2.5 5.0 7.5 10.0 1
0
4
Tensile Str
91
Figure 4.6 Correlation between dynamic tensile strength and elastic modulus
0 4 8 12 16 20 24 28 32 36 40
0
4
8
12
16
D
y
n
a
m
i
c

t
e
n
Elastic modulus (GPa)
40
20
24
28
32
36
s
i
l
e

s
t
r
e
n
g
t
h

(
M
P
a
)
Reduction ratio 1
Reduction ratio 2
0
36
n
s
Reduction ratio 1
40
20
24
28
32
i
l
e

s
t
r
e
n
g
t
h

(
M
P
a
)
Reduction ratio 2
4
8
12
16
D
y
n
a
m
i
c

t
e
0 25 50 75 100 125 150 175 200 225 250
Compressive strength (MPa)
Figure 4.7 Correlation between dynamic tensile strength and compressive strength

92

4.2.2 SPECIFIC COMMINUTION ENERGY

The DATA 6000 waveform analyzer developed a power curve for each test specimen
using the stroke and force input signals. The area under the power curve, initiating at the
point of first fracture and ending at the moment the actuator began to retract, was
calculated by the DATA 6000 in order to determine the net crushing energy. The net
crushing energy divided by the mass of the specimen is the specific comminution energy,
the energy required to reduce the size of the particle for the given operating conditions (in
kilowatt-hours per metric ton). Table 4.3 lists the specific comminution energy results
for each rock at each reduction ratio. Individual specimen data can be found in Appendix
II.

Figures 4.9 through 4.15 are sample HECT plots representing typical/average
behavior for each rock at each reduction ratio. The visual quality of the plots is poor and
the velocity curve developed by the DATA 6000 is quite ragged. This is a result of
differentiation of the digitally recorded stroke signal processed from an analog input
signal. Despite the ragged appearance there is little to no difference between the digitally
processed velocity curve and one developed from direct differentiation of the analog
signal (Allis-Chalmers, 1985). Also note that for the stroke signal (BUF.A1) one volt is
equivalent to 10.16 mm and for the load signal (BUF.A2) one volt is equivalent to 15.57
kN.
Figure 4.8 Correlation between dynamic tensile strength and fracture toughness
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0
4
8
12
16
D
y
n
a
m
i
c

t
e
n
s
K
Ic
(MPa m
0.5
)
20
24
28
32
36
40
e

s
t
r
e
n
g
t
h

(
M
P
a
)
Reduction ratio 1
Reduction ratio 2
i
l
93

Table 4.3 Specific comminution energy results from the HECT
Reduction Ratio 1 Reduction Ratio 2
E
c
E
c
Rock Type
kWh/t kWh/t
Shadwell Metabasalt 0.407 0.550
Boscobel Granite 0.105 0.230
Culpeper Grey Siltstone 0.292 0.350
Culpeper Red Siltstone 0.298 0.320
Leesburg Diabase 0.371 0.477
Thornburg Granite 0.272 0.384
Spotsylvania Granite 0.233 0.356

Figures 4.9 through 4.15 show the behavior of the rocks under jaw crusher conditions.
At the start of the crushing cycle the actuator, which represents the swing jaw, moves
towards the particle following the predefined haversine path. When the actuator comes in
contact with the particle there is a sharp increase in the load and a corresponding increase
in power consumption. Failure of the particle occurs quite rapidly (within approximately
1 s) and is followed by a sharp decrease in load. The power also drops off after failure.
Since particle failure requires only small amount of strain input the actuator continues to
move upward (i.e., the crusher chamber continues to close). Fragmented particles
remaining inside the crushing chamber (Figure 3.13) with dimensions greater then the
closed side set are crushed under the continued displacement of the actuator. This is
depicted by post-failure increases in the load and power signals. After traveling the
throw distance the actuator begins to retract and particle crushing ends.

The HECT plots along with the results in Table 4.3 reveal that the rocks that required
the most energy to crush also experienced the most secondary breakage. Secondary
breakage is the crushing of daughter fragments after particle failure. The Culpeper
siltstones exhibited the smallest amounts of secondary breakage. Even the Boscobel
granite, which on average required less energy to crush than the siltstones, underwent
more secondary crushing, i.e., a larger percentage of the total specific comminution
energy was a result of secondary breakage. The siltstones required more energy to
initiate fracture due to their high strength, low elastic modulus behavior, and thus more of
the actuator displacement was used during particle failure. In comparison to the other
rocks the Culpeper siltstones take approximately 43% to 94% more strain to fail (based
on tensile strength and elastic modulus).

The strain at failure is very small in comparison to the total displacement of the
actuator and a more significant explanation for the siltstones behavior is needed. During
testing it was observed that the siltstones tended to shatter producing thin, flat daughter
fragments. This was a result of ideal disk splitting. Under diametrical compression
94
Figure 4.9 Typical BG HECT behavior: CSS 1 (top) and CSS 2 (bottom)
95
Figure 4.10 Typical CRS HECT behavior: CSS 1 (top) and CSS 2 (bottom)
96
Figure 4.11 Typical CGS HECT behavior: CSS 1 (top) and CSS 2 (bottom)
97
Figure 4.12 Typical LD HECT behavior: CSS 1 (top) and CSS 2 (bottom)
98
Figure 4.13 Typical SG HECT behavior: CSS 1 (top) and CSS 2 (bottom)
99
Figure 4.14 Typical SMB HECT behavior: CSS 1 (top) and CSS 2 (bottom)
100

Figure 4.15 Typical TG HECT behavior: CSS 1 (top) and CSS 2 (bottom)
101
102
energy required for fracture is dependent upon, among other things, the stress applied to
the particle, the strength of these correlations is not unexpected. More significantly, the
correl
fracture initiates at the center of a disk and typically propagates in a straight line towards
the load contact points. Additional displacement results in fracture planes on each side
and parallel to the first plane. The result is rectangular plate-like fragments. Under rapid
displacement rates of the HECT system these fragments move outward and away from
the contact points at velocities greater than the velocity of the actuator. The daughter
fragments of the Culpeper rocks either exited the crushing chamber or fell in such a way
that their relative thinness, in comparison to the CSS, prevented any further crushing.
Crack propagation in the other rocks, based on observations from fracture toughness
testing, most likely did not occur in a straight path, resulting in larger and more
irregularly shaped daughter fragments that were subject to additional breakage. These
rocks experienced more cleavage fracture, particularly at the lower reduction ratio. This
also implies that shatter and cleavage are not only functions of the type of crushing
machine used but also of the fracture behavior of the material.

The increase in reduction ratio resulted in an increase in specific comminution
energy. The increase in energy was due to the larger amount of secondary breakage that
occurs at larger reduction ratios. Since the energy required for initial fracture essentially
did not vary between reduction ratios (assuming load at first fracture and dynamic tensile
strength are representative of the energy required to initiate fracture), the amount of
crushing that occurs after initial breakage is dependent upon the post-failure strain input
into the particle. A decrease in the closed side set of the HECT system results in more
post-peak displacement and significant secondary crushing of daughter fragments.

Although an energy increase due to an increase in reduction ratio was seen for each
rock changes in crusher reduction ratio (or CSS) cannot be expected to have a universal
effect on all rocks. Table 4.3 indicates that the Culpeper siltstones saw the smallest
increase in specific comminution energy, an average of 0.040 kWh/t. The other five
rocks exhibited an increase that ranged between 0.106 - 0.143 kWh/t (average of 0.122
kWh/t). The dissimilar behavior, or variance in the degree of behavior, of the Culpeper
siltstone is a result of the mechanical and fracture properties of the rock. Again the
shattering behavior shown by the siltstones resulted in the exiting of daughter fragments
from the crushing chamber. More secondary breakage of the siltstone is only possible if
the crusher speed is increased such that the actuator is able to close in and make contact
with daughter fragments before they exit the crushing chamber.

The specific comminution energy results were compared to the mechanical properties
of the rocks, as well as the fracture toughness results and dynamic tensile strength results.
Figures 4.16 - 4.20 show the correlations between E
c
and these properties for each
reduction ratio. The strongest correlations are between E
c
and tensile strength and E
c
and
fracture toughness. Since tensile strength and mode I fracture toughness are most
representative of the stress situation within a particle under point contact loading and the
E
c
. ation between E
c
and K
Ic
is stronger than that between tensile strength and
103
ratio are made. The energy required to crush rock material is a clearly function of its
resistance to fracture and the amount of strain input by the crushing machine.
Assessing the fracture resistance of a rock particle in terms of a property that relates the
fracture strength of the particle to the presence of flaws within the particle normalizes the
effect and variation of stress magnitude. Tensile strength is an ultimate strength criterion
and is based strictly on the magnitude of stress that a particle can withstand. It is very
applicable to situations where tensile stresses lead to fracture by opening cracks and
flaws but it is subject to more variation since no account is made for the presence or
geometry of those flaws.

A limiting relationship between E
c
and elastic modulus is evident in Figure 4.17, with
a stronger correlation for the larger reduction ratio. The specific comminution energy
seems to increase exponentially with elastic modulus, although more very stiff rocks (E >
28 GPa) would need to be tested in order to suggest such a relationship. Some
correlation between specific comminution energy and compressive strength is indicated
in Figure 4.18 for reduction ratio 1. The correlation loses its power significantly with an
increase in reduction ratio. Compressive strength most likely correlates fairly well with
E
c
at low reduction ratios where power consumption is more dependent upon the material
strength. At higher reduction ratios the specific comminution energy is more dependent
on the fracture and elastic properties, i.e., fracture strength, crack propagation, and strain
energy release rate. Figure 4.20 illustrates the relationship between E
c
and dynamic
tensile strength, which is more scattered than expected due to the low dynamic tensile
strength of the Shadwell metabasalt.

The relationship between E
c
and K
Ic
in Figure 4.19 may be skewed somewhat by the
results of the Edge Notched Disk test. It is widely accepted that rock fracture toughness
measurements and values show a dependence on grain size. This dependency is evident
in the fracture toughness values presented here and was discussed in section 4.1. The
data presented in Figure 4.19 show that the Culpeper siltstones have the highest fracture
toughness values yet the corresponding specific comminution energy values for those
rocks fall in the middle to bottom third of the overall results. In fact the relationship
between E
c
and K
Ic
is essentially linear except for the Culpeper siltstones. Based on the
grain size effect, the fracture toughness values of the other, larger grained rocks may be
underestimated, resulting in a left-shift of their data. If this is the case the correlation
between E
c
and K
Ic
may in fact be stronger than is demonstrated.

The most important aspect of the HECT specific comminution energy results is that
fracture toughness correlates the strongest with E
c
in comparison to other rock properties.
Based on the physics of fracture and the fact that the energy required to fracture a particle
is dependent on the stresses applied to it, its mechanical properties, and the presence of
flaws, it is no revelation that Mode I fracture toughness is related to the energy required
to reduce the size of a particle under point contact loading. Another important result is
that the relationship between E
c
and K
Ic
remains evident even when changes in reduction
104
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 22.5 25.0
0.00
0.06
0.12
0.18
0.24
0.30
0.36
0.42
0.48
0.54
0.60
E
c

(
k
W
h
/
t
)
Tensile strength (MPa)
Reduction ratio 1
Reduction ratio 2
Figure 4.16 Correlation between E
c
and tensile strength
Figure 4.17 Correlation between E
c
and elastic modulus
0 4 8 12 16 20 24 28 32 36 40
0.00
0.06
0.12
0.18
0.24
0.30
0.36
0.42
0.48
0.54
0.60
E
c

(
k
W
h
/
t
)
Elastic modulus (GPa)
Reduction ratio 1
Reduction ratio 2
Figure 4.18 Correlation between E
c
and compressive strength
0 25 50 75 100 125 150 175 200 225 250
0.00
0.06
0.12
Compressive strength (MPa)
0.18
0.24
0.30
0.36
0.42
0.48
0.54
0.60
E
c

(
k
W
h
/
t
)
Reduction ratio 1
Reduction ratio 2
Figure 4.19 Correlation between E
c
and fracture toughness
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0.00
0.06
0.12
0.18
K
Ic
(MPa m
0.5
)
0.60
0.24
0.30
0.36
0.42
0.48
0.54
E
c

(
k
W
h
/
t
)
Reduction ratio 1
Reduction ratio 2
105
0 3 6 9 12 15 18 21 24 27 30
0.00
0.06
0.12
0.18
0.24
0.30
E
c

(
k
W
h
/
t
)
Dyna

4.2.3 BREAKAGE FUNCTION

After single particle breakage was conducted the pooled samples for each rock were
sieved in order to determine the fragmented size distribution. A 2 sieve series was used
starting with an initial opening size of 38.1 mm and finishing with a 0.075 mm mesh
opening size. The breakage distribution was determined by normalizing the fragmented
sizes with the parent size. The parent size was taken to be the average diameter of the
tested specimens. Appendix III contains the fragmented size distribution data.

he breakage functions for each rock, and for both reduction ratios, are displayed in
Fig
ccurs in an
actual crusher. If the irregular particles crushed in actual machines are idealized as
T
ures 4.21 and 4.22. In each instance all material passed at a fragmented size of
approximately 0.8 of the parent size. In fact for each rock 100% of the fragmented
material would have passed at a size close to 0.54 of the parent size. This is because disk
specimens with thicknesses approximately one half of their diameter were used in this
study. Under (small displacement) point contact loading, a disk, with dimensions DDt,
tends to split in half and the resultant fragments have approximate dimensions of
D0.5Dt. When the thickness is close to the radius of the disk, 100% of the broken
fragments are retained on a sieve that has an opening equal to one-half of the parent size.
There was some debate as to whether or not this behavior is typical of what o
0.36
0.42
0.48
0.54
0.60
Reduction ratio 1
Reduction ratio 2
Figure 4.20 Correlation between E
c
and dynamic tensile strength
mic tensile strength (MPa)
106
107
spheres, then disk specimens with thicknesses near one half of their diameters are also
representative of irregular particles. Small displacement, point contact loading of a
sphere will result in four lune pieces with dimensions D0.5D0.5D, which also results
in 100% of the fragmented pieces being retained on a sieve that has an opening equal to
one-half of the parent size. Of course a sphere, or an equivalent disk, may not adequately
describe irregular particles so the results of this study may be limited/effected by the
chosen test specimen.

For the smaller reduction ratio (Figure 4.21) it can be seen that the Boscobel,
Shadwell, Leesburg, Spotsylvania, and Thornburg rocks exhibit similar distributions
between the parent size and the 0.1 size. At sizes smaller than 1/10
th
of the parent size
the distributions diverge. At larger fragment sizes these rocks experience approximately
the same degree of cleavage but the proportion of material in the smaller size ranges
varies according to the degree of shatter each rock is subjected to. Figure 4.21 also
touches upon the observation made in the previous section that fracturing of the Culpeper
siltstones was a shatter-dominated event. The siltstones are shown to have a uniform
distribution and their breakage functions are not a mixture of separate size populations.
The other five rocks exhibit multi-modal behavior and their breakage functions are the
more typical m .
Figure 4.21 Breakage functions for each rock at reduction ratio 1
0.0 1
0.01
0.1
1
1 0.1
B

(
d
i
,
d
1
)
d
i
/d
1
BG
CGS
CRS
LD
SMB
SG
TG
ixture of separate populations caused by shatter and cleavage
108

Increasing the reduction ratio resulted in a breakage distributions consisting of a
larger proportion of fine progeny sizes (i.e., the percent passing each progeny size
increased). Figure 4.22 indicates that the distribution modulus of the larger progeny size
populations is approximately the same for each rock but the proportion of fragmented
material in the larger sizes is not. There is no consistency among the rocks distribution
modulus or material proportion in the finer sizes (less than 1/10
th
the parent size). The
Culpeper siltstones again yielded fairly uniform breakage functions. The siltstones also
did not experience as great of an increase in the percentage passing each fragment size.
This is most likely a result of the small degree, relative to the other rocks, of secondary
crushing the siltstones were exposed to and corresponds to the proposition made in the
previous section that an increase in reduction ratio does not affect the siltstones breakage
behavior as much as the other rocks.
Figures 4.23 4.29 are the separate breakage functions for each rock at each
reduction ratio. The change in breakage distribution with reduction ratio can be seen
more clearly for each rock. At the lower reduction ratio the breakage functions are a
mixture of two separate size populations, except for the Culpeper rocks as noted earlier.
At the higher reduction ratio the larger size population becomes bimodal due to increased
secondary bre
Figure 4.22 Breakage functions for each rock at reduction ratio 2
0.01 0.1 1
0.01
0.1
1
B

(
d
i
,
d
1
)
d
i
/d
1
BG
CGS
CRS
LD
SMB
SG
TG
akage and shattering of progeny fragments.
0.01 0.1 1
0.01
0.1
1
Reduction ratio 1
Figure 4.23 Breakage functions for Boscobel granite
Reduction ratio 2
B

(
d
i
,
d
1
)
d
i
/d
1
0.01 0.1 1
0.01
B

(
d
d
i
/d
1
Figure 4.24 Breakage functions for Culpeper gray siltstone
0.1
1
Reduction ratio 1
Reduction ratio 2
i
,
d
1
)
109
Figure 4.25 Breakage functions for Culpeper red siltstone
0.01 0.1 1
0.01
0.1
1
Reduction ratio 1
Reduction ratio 2
B

(
d
i
,
d
1
)
d
i
/d
1
0.1
Reduction ratio 2
i
,
d
1
)
0.01 0.1 1
0.01
1
B

(
d
d
i
/d
1
Figure 4.26 Breakage functions for Leesburg diabase
Reduction ratio 1
110

Figure 4.27 Breakage functions for Spotsylvania granite
0.01 0.1 1
0.01
0.1
B

(
d
i
,
d
1
)
1
Reduction ratio 1
Reduction ratio 2
d
i
/d
1
Figure 4.28 Breakage functions for Shadwell metabasalt
0.01 0.1 1
0.01
d
i
/d
1
0.1
1
Reduction ratio 1
Reduction ratio 2
B

(
d
i
,
d
1
)
111
Conventional wisdom, based on prior research (see section 2.4), dictates that an
increase in energy intensity changes only the proportion of the separate populations that
make up a breakage function and not their distribution. Agreement with this perception
is found in the preceding figures. The distribution of the finer size population (sizes
below 1/10
th
of the parent size) does not change but the proportion of material in that
population does change. The proportion of material in the larger size fractions decreases
because of rebreakage of cleavage progeny fragments. The distribution modulus does not
change, although a change in the distribution above sizes half the parent size can be seen
in some instances. This indicates that rebreakage of daughter particles initially produced
by cleavage is occurring (note that the slope/distribution modulus above 0.5 is
approximately the same as that of the fine size population).

In section 2.4.3 the t
10
parameter was introduced. It is the percent of progeny
particles passing a size one-tenth of the initial original particle size. t
10
is employed as a
characteristic size reduction parameter and has been related to the energy absorbed during
a single particle breakage test. For crushing applications t
10
is usually in the range of
10% to 20%. In order to determine t
10
, and other t
n
values, from the breakage distribution
data, a MathCAD program was developed so that cubic spline interpolation of the
measured data could be performed. Table 4.4 lists the value of t
10
for each rock at each
reduction ratio. As expected t
10
increased with an increase in reduction ratio and the
results fit close to the range of values typical of crushing. In conjunction with Table 4.3
Figure 4.29 Breakage functions for Thornburg granite
0.01 0.1 1
0.01
0.1
1
Reduction ratio 1
Reduction ratio 2
B

(
d
i
,
d
1
)
d
i
/d
1
112
113
the results listed below show that the rocks that had the largest percentage increase in E
c

also had the largest percentage increase in t
10
.

Table 4.4 Values of t
10
for each rock and reduction ratio
Reduction Ratio 1 Reduction Ratio 2
t
10
t
10
Rock Type
% %
Shadwell Metabasalt 8.1 15.0
Boscobel Granite 7.4 17.0
Culpeper Grey Siltstone 11.7 16.8
Culpeper Red Siltstone 12.9 12.8
Leesburg Diabase 8.7 15.8
Thornburg Granite 8.7 15.7
Spotsylvania Granite 8.5 20.4

For each specific rock the t
10
parameter is a function of the energy input. On the
whole, looking at the grouped results for all rocks, t
10
is more so a function of the
reduction ratio. Figure 4.30 shows that t
10
increases with an increase in reduction ratio
but that at each reduction ratio it does not change significantly as E
c
increases. This is
because at a given reduction ratio specific comminution energy is a function of the
material behavi ws that t
10
varies
Figure 4.30 Relationship between t
10 c
and E
0.00 0.06 0.12 0.18 0.24 0.30 0.36 0.42 0.48 0.54 0.60
0.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
22.5
25.0
t
1
0

(
%
)
Specific comminution energy (kWh/t)
Reduction ratio 1
Reduction ratio 2
or. Thus a plot of t
10
versus fracture toughness also sho
114
very little as K
Ic
changes, although the change in t
10
with an increase in reduction ratio is
still evident (Figure 4.31). Figures 4.32 and 4.33 indicate that other t
n
values show
similar behavior when compared to fracture toughness, particularly smaller values. t
2
and
t
4
show a fairly wide degree of variation as fracture toughness changes. t
25
, t
50
, and t
75

fluctuate in a manner similar to t
10
.

It is difficult to observe any noticeable trend or correlation between t
n
values and K
Ic
.
In regards to t
n
and K
Ic
, the conventional thinking would be that as fracture toughness
increases the percent passing each t
n
value decreases, or tougher rocks would fracture
more coarsely. However, high fracture toughness rocks require more strain energy input
in order to propagate flaws and thus tend to shatter upon fracture, which produces finer
sized fragments. The siltstone is an example of this behavior and in Figures 4.32 and
4.33 it can be seen that more material passes at t
2
and t
4
for the tough siltstones. The
relationship between t
n
and K
Ic
is complicated further by grain size. Rock fracture
toughness has been shown to be dependent upon grain size (in this study and in the
literature), as large grained rocks tend to have lower K
Ic
values. Subsequently, on the one
hand, large grained rocks are easy to fracture and should produce a more finely sized
breakage distribution based on their fracture toughness. On the other hand, large grained
rocks tend to fracture along grain boundaries and progeny fragment size will be
dependent on the grain size and may end up being coarser in comparison to tougher, fine-
grained rocks. The Boscobel granite is a good example of this behavior. In Figures 4.32
Figure 4.31 Relationship between t
10
and K
Ic

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
22.5
25.0
t
1
0

(
%
)
Fracture toughness (MPa m
0.5
)
Reduction ratio 1
Reduction ratio 2
and 4.33 the Boscobel granite, which has the lowest fracture toughness, has the highest
percentage passing at t
2
and t
4
. This indicates that multiple crack surfaces, occurring as
grain boundaries, are propagated upon fracture. However, at lower t
n
values the Boscobel
has a smaller percentage passing. These sizes are near the grain size of the granite and
the progeny fragments at these sizes were most likely produced at first fracture or soon
there after. In other words, large grained rocks shatter more upon initial fracture
producing daughter fragments that are not subjected to secondary breakage. In order to
determine a relationship between fracture toughness and breakage function, as
represented by a series of t
n
values, observation of the fracture behavior and
structure/grain size of each rock is required.

A final observation from the breakage distribution results is that the t
10
parameter
measured under jaw crusher conditions can be applied in order to determine a complete
size distribution. This is the one-parameter family of curves method described in section
2.4.3, where, using only a given value of t
10
, the full product size distribution can be
reconstructed. Figure 4.34 shows the HECT breakage results in terms of t
n
versus t
10
.
The solid lines represent Narayanans (1985) work presented previously in Figure 2.21.
Narayanans work was for tumbling mills and there seems to be some agreement, at least
trend-wise, between that work and the results from this study. The t
n
values from the
HECT jaw crusher set-up are higher than those given by Narayanan, most likely because
of the different size reduction application (jaw crusher vs. tumbling mill), different
degrees of reduction ratio, and the fact that Narayanans data represents only one type of
material, where as the data in this study covers six different rock types (although
Naraynans relationships have since been verified for a range of rock/ore types). Despite
these differences it is clear that t
10
is mostly a function of reduction ratio, and not
necessarily of energy input or size reduction machine, and that if t
10
and fracture
toughness can be related, then all other t
n
values can be determined simply from t
10
.
115
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0
10
20
30
40
50
60
70
80
90
100
t
2
t
4
t
10
t
25
t
50
t
75

Figure 4 ratio 1 .32 Relationship between various t
n
and K
Ic
for reduction
Ic
t
n

(
%
)
K (MPa m
0.5
)
Figure 4.33 Relationship between various t
n
and K
Ic
for reduction ratio 2
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0
10
20
30
K
Ic
(MPa m
0.5
)
40
50
60
70
80
90
100
t
n

(
%
)
t
2
t
4
t
10
t
25
t
50
t
75
116

Figure 4.34 t
n
vs. t
10
overlapped with Narayanans one-parameter family of curves
0 10 20 30 40 50
0
20
40
80
100
60
t
n

(
%

p
a
s
s
i
n
g
)
t
t
75
t
50
t
25
t
10
t
4
t
2 t
2
t
4
t
10
t
25
50
t
75
t
10
(%)
117
CHAPTER 5. MODEL DEVELOPMENT AND EXPERIMENTAL VERIFICATION


5.1 MODELS FOR JAW CRUSHER POWER CONSUMPTION AND PRODUCT SIZE

5.1.1 POWER CONSUMPTION

The results presented in section 4.1 indicated a strong correlation between fracture
toughness, K
Ic
, and specific comminution energy, E
c
. This relationship can be used to
develop an empirical model for the prediction of jaw crusher power consumption that can
account for changes in reduction ratio.

As was noted earlier the Culpeper gray and red siltstones were tested separately but
when no statistical difference was found between their mechanical properties or fracture
toughness values it was decided that they would be considered as one rock. Therefore the
HECT results for the gray and red siltstone have been pooled together and model
development is based only on data for six rock types.

Figure 5.1 is a plot of specific comminution energy versus fracture toughness for
reduction ratio 1. It displays a best-fit line of the data along with the upper and lower
95% confidence limits of the correlation. The linear fit has been forced through the
origin based on the assumption that a material with no fracture toughness would not
require energy in order to be fractured. The linear relationship between E
c
and K
Ic
is
given by the following equation:
[5.1]
The coefficient of determination for the relationship given in Equation 5.1 is 0.89. When
the reduction ratio is so changes. This is
en in Figure 5.2. The same type of relationship holds, with E
c
and K
Ic
being linearly
rela
0.255 {for a reduction ratio of 1.50}
c Ic
E K =
increased the relationship between E
c
and K
Ic
al
se
ted. The relationship can be expressed by:
0.342 {for a reduction ratio of 2.97}
c Ic
E K = [5.2]
The coefficient of determination for the relationship between E
c
and K
Ic
for reduction
ratio 2 is 0.80. An increase in reduction ratio results in an increase in the slope, m, of
Equation 5.1. In order to determine a model that is inclusive of all reduction ratios an
expression relating the slopes of Equations 5.1 and 5.2 needs to be determined.

In addition to the two reduction ratios used in this study, it can be assumed that at a
reduction ratio of one (i.e, the particle size equals the closed side set and no strain energy
is input into the particle), the slope of the relationship between E
c
and K
Ic
is also zero. A
plot of this data is given in Figure 5.3 and has been fit with two expressions, one for
reduction ratios between 1 and 1.5 and the other for reduction ratios greater than or equal
to 1.5 . The general relationship between E
c
and K
Ic
, covering all reduction ratios (RR), is
then given by the following equation:
118
119

Figure 5.1 Linear fit of E
c
and K
Ic
data for reduction ratio 1
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0.00
0.05
0.10
0.15
0.20
E
K
Ic
(MPa m
0.5
)
0.48
0.54
0.60
Linear fit for reduction ratio 2
Upper 95% Co fidence Limit n
Lower 95% Confidence Limit
E
c

(
k
W
h
/
t
)
K
Ic
(MPa m
0.5
)
c

(
k
0.25
0.30
0.35
0.40
0.45
0.50
Linear fit for reduction ratio 1
Upper 95% Confidence Limit
Lower 95% Confidence Limit
W
h
/
t
)
Figure 5.2 Linear fit of E
c
and K
Ic
data for reduction ratio 2
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0.00
0.06
0.12
0.18
0.24
0.30
0.36
0.42
120

| |
0.428
0.511 0.511 for 1 1.5
0.215 for 1.5
c Ic
c Ic
E RR K
E RR K RR
= + <
( =

RR
[5.3]

An allometric equation was chosen for reduction ratios greater than or equal to 1.5.
Although the measured data only suggest a linear relationship between reduction ratios of
1.5 and 2.97 it has been assumed that a limiting power law, or possibly exponential,
relationship between specific communution energy and reduction ratio exists at higher
reduction ratios. This is mainly due to the fact that the rate of strain input decreases with
increasing reduction ratio.

Equation 5.3 governs the relationship between fracture toughness and specific
comminution energy based on the HECT results, where E
c
is given in terms of kilowatt-
hours per metric ton and reduction ratio is defined as the particle size divided by the
closed side set. In order to determine the power consumption of a jaw crusher the feed
size distribition would have to be known, as well as the mass flow at each size, and the
the idling power of the crusher. Equation 5.3 can be adjusted to account for these
variables and an equation for determining the power consumption of a jaw crusher, based
on fracture toughness, can be written as:
Figure 5.3 Change in coefficient m with reduction ratio
0 1 2 3 4 5 6 7 8 9 10
0.00
0.06
0.12
0.18
0.24
0.30
0.36
0.42
0.48
0.54
0.60
m = -0.511+0.511RR
m = 0.215RR
0.428
S
l
o
p
e
,

m
Reduction ratio, RR
| |
1
0.428
1
0.511 0.511 for 1 1.5
0.215 for 1.5
j
c i Ic i i n
i
j
c i Ic i i n
i
P RR K C x P
P RR K C x P RR
=
=
= + + <
(
= +

RR

[5.4]
where, P
c
is the power consumption of the crusher in kW
RR
i
is the reduction ratio for a particle of size i
C
i
is the probability of breakage for particle size i
x
i
is the mass flow of particle size i in metric tons per hour
P
n
is the power drawn by the crusher under no load in kW.
Equation 5.4 accounts for the wide range of reduction ratios seen in a jaw crusher due to
the non-uniformity of the feed size distribution. It is difficult in practice to sample the
feed size entering a jaw crusher and often times the feed size distribution is unknown. In
these instances Equation 5.4 is not applicable. If an estimate of the average reduction
seen by the feed can be made, which infers some kowledge of the feed size distribution,
then the following equation can be used to approximate the power consumption of a jaw
crusher:

RR

[5.5]
w
T is the total mass flow in metric tons per hour.

Equation 5.4 is limited since the E
c
and K
Ic
relationship is based on six rocks and two
reduction ratios. Figures 5.1 and 5.2 indicate that the relationship may be exponential or
allometric, with a limiting amount of specific comminution energy required to crush high
fracture toughness rocks (i.e., a ceiling is reached in terms of E
c
). The strength of the
linear relationships suggested in Equations 5.1 and 5.2 is influenced by the behavior of
the Culpeper siltstone, which at each reduction ratio has E
c
values outside the 95%
confidence limit of the linear fit model. As discussed in section 4.2, the siltstone was not
exposed to as much secondary breakage as the other rocks. This is mainly due to the low
elastic modulus of the siltstone. Though the siltstone was the only tough rock that
exhibited this type of behavior, it is believed that E
c
also has a strong dependency on the
elastic modulus and over the course of more testing, linear, or non-linear, relationships
between E
c
and K
Ic
may be strengthened if they are defined for specific intervals of
elastic modulus. Another option may be to use the strain energy release rate, G, which is
dependent on fracture toughness and the elastic modulus (and in the case of plane strain,
Poissons ratio). G, or an estimate of G based on the measured K
Ic
and elastic modulus
values for each rock, was investigated for this study but the correlation, and the strength
of the relationship, between strain energy release rate and E
c
was not as significant as the
one between E
c
and K
Ic
.

0.428
0.511 0.511 for 1 1.5
0.215 for 1.5
c avg Ic n
c avg Ic n
P RR K T P
P RR K T P RR
( = + + <

(
= +

here, RR
avg
is the avereage reduction ratio for the entire feed
121
5.1.2 PRODUCT SIZE

A direct model for the product size appearing from a jaw crusher is not possible. Any
n, and the
reakage function. The model developed here is strictly for the prediction of the
onjunction with the feed size distribution and classification
nction can be used to find the product size distribution from a jaw crusher.
the one-parameter family of curves approach
opularized by the JKMRC (Julius Kruttschnitt Mineral Research Center). Each method
age functions (2 for each rock) was
rduous.

these relationships. Thus the one-parameter family of
curves can be used to determine the breakage function for a ro
ults are pooled and presented as one rock
because the percent passing at a particle size one-tenth the parent si
e weaker
cks that are activated upon fracture and produce progeny fragments that are not
sly. This process is shatter fracture and results in
prediction of product size is dependent upon the feed size, classification functio
b
breakage function, which in c
fu

In section 2.4.3 methods for describing the breakage function were covered. The
most common method is to express the breakage function as a mixture of separate size
populations. A second method is
p
was given due consideration for the breakage data collected in this study but the final
model used here employs an approach similar to that of the one parameter family of
curves. For the mixed population approach, relationships between the coefficients K
i
and
n
i
, used to describe different size populations, and fracture toughness were difficult to
discern and curve fitting of the twelve measured break
a
The model developed here is based on the characteristic size reduction parameter t
10
.
In section 4.2.3 a plot, inclusive of all reduction ratios, of t
n
values versus t
10
was
presented that showed relationships exist between different t
n
values and t
10
. It was
determined that if t
10
is a function of fracture toughness and reduction ratio, then other t
n

values can be determined using
ck given the fracture
toughness and reduction ratio.

Figure 4.31 in section 4.2.3 showed the relationship between t
10
and fracture
toughness at each reduction ratio. A correlation between the two is hard to observe, even
when the siltstone res . This is most likely
ze, or any size for that
matter, is dependent upon the reduction ratio, the rocks fracture behavior, and the rocks
grain size/structure. The data presented in section 2.4.3 indicated that at small t
n
values
the percent passing increases with increasing fracture toughness for the smaller reduction
ratio. One explanation for this behavior is the increased presence of flaws in th
ro
subjected to secondary breakage. This corresponds with the behavior observed at the
higher reduction ratio that showed a decrease in percent passing with increased fracture
toughness at the small t
n
values. The effect of grain size/structure is nullified when
significant amounts of secondary breakage occur. A second explanation is based on the
ability of the rock particle to absorb energy. It is possible that before fracture occurs at
the weakest flaw in a rock particle, excess energy can be applied such that fracture of
many flaws occurs almost simultaneou
122
finer progeny fragments. Again, at higher reduction ratios this behavior is hidden due
large amounts of secondary breakage.

Based on the previous qualitative description, the effect of fracture behavior and
microstructure on the breakage distribution should be accounted for when investigating
the relationship between t
10
and K
Ic
. The only approach that can be taken here is to
account for a rock particles ability to absorb energy by estimating the strain energy
release rate, G, of each rock. Recalling Equation 2.34, G can be calculated using fracture
toughness, elastic modulus, and Poissons ratio. For conditions of plane stress, only K
Ic

and the elastic modulus are needed. Poissons ratio for the rocks used in this study are
not known thus a universal value of 0.3 was assumed since Poissons ratios for rocks are
fairly small and usually very comparable. Table 5.1 lists the G values for the rocks used
in this study, along with their K
Ic
values for reference.

Table 5.1 Strain energy release rate of each rock
G K
Ic

Rock Type
2
J m MPa m
Shadwell Metabasalt 60.15 1.411
Boscobel Granite 15.98 0.601
Culpeper Siltstone 79.54 1.477
Lees . burg Diabase 47 95 1.254
Thornburg Granite 36.36 1.045
Spotsylvania Granite 25.22 0.843

Figures 5.4 and 5.5 are plots of t
10
versus G for each reduction ratio. An allometric
expression has been fit to the data in each case. For a reduction ratio of 1.50 t
10
can be
determined from:

0.251
10
3.54
'
Ic
K
t
E
| |
=
|
\ .
[5.6]
where E is the elastic modulus. For a reduction ratio of 2.97 the relationship between t
10

and K
Ic
can be written as:

0.139
10
27.2
'
Ic
K
t
E

| |
=
|
\ .
[5.7]

It is clear that the relationship between t
10
and K
Ic
2
/E is not well defined but the
coefficient of determination is higher than that for any relationship defined between t
10

and K
Ic
. The relationship shown in Figure 5.4 demonstrates that t
10
increases with G.
This is because for rocks with higher G values shatter fracture is more prevalent and
more energy is released as new crack surface energy. Figure 5.5 indicates that as the
reduction ratio is increased, the relationship between t
10
and G inverses, and t
10
decreases
with increasing strain energy release rate. The low G rocks are subjected to more
123

Fi
10
1 gure 5.4 Allometric fit of t versus G for reduction ratio
0 10 20 30 40 50 60 70 80 90 100
0.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
22.5
25.0
t
1
0

(
%

p
a
s
s
i
n
g
)
G (J/m
2
)
Allometric fit for reduction ratio 1
Figure 5.5 Allometric fit of t
10
versus G for reduction ratio 2
0 10 20 30 40 50 60 70 80 90 100
22.5
25.0
12.5
15.0
17.5
20.0

p
a
s
s
i
n
g
)
0.0
2.5
5.0
7.5
10.0
t
1
0
Allometric fit for reduction ratio 2

(
%
G (J/m
2
)
124
secondary breakage than the higher G rocks because the low G rocks initially produce
large progeny fragments as a result of cleavage. These progeny undergo additional
breakage in comparison to the first fragments of the high G rocks, prod
and having more particles of varying size.

7 are functions of the reduction
expressions relating them to the
a, is plotted against reduction
1 means that no size
d and all t
n
s are zero.
e functions used to relate the coefficient a and the reduction ratio. Two
expressions were used, one for reduction ratios between 1 and 1.5, and one for reduction
ratios greater than or equal to 1.5.
The exponent, defined as b, in Equations 5.6 and 5.7 is more difficult to relate to the
reduction ratio. In this case b decreases and becomes negative between the reduction
uced by shatter
The coefficients and exponents in Equations 5.6 and 5.
ratio. Thus, as was done for the E
c
model, general
reduction ratio need to be determined. The coefficient,
ratio in Figure 5.6. As was noted previously a reduction ratio of
reduction takes place, therefore no daughter fragments are produce
Again, since no reduction ratios greater than 3 were used some assumptions have to be
made about what occurs as the reduction ratio increases. In this instance it is known that
t
10
cannot exceed 100%, thus a has an upper limit as the reduction ratio increases. It is
expected that a will increase at about the same rate as defined by the two measured data
points up to a reduction ratios of about 5. Above a reduction ratio of 5, a will not
increase as rapidly and will approach 100 at a reduction ratios greater than 10. Figure 5.6
shows th

Figure 5.6 Change in coefficient a with reduction ratio
0 1 2 3 4 5 6 7 8 9 10
0
10
20
30
40
50
60
70
80
a = 100 - 100/(1+(RR/3.97)
3.40
)
a = -7.095 + 7.905RR
C
o
e
f
f
i
c
i
e
n
t
,

a
Reduction ratio
90
100
125
ratios of 1.5 and 2.97. But for a reduction ratio of one b is zero, as is a, and at some large
duction ratio b equals zero again (in order to satisfy the t
10
upper limit of 100%). For re
reduction ratios between 1 and 1.5 the change in b is assumed linear. For reduction ratios
greater than 1.5 b decreases in a exponential/linear manner up to a reduction ratio of 3.
Beyond a reduction ratio of 3 b increases and approaches zero in an exponential/linear
manner. The functions governing the change in b with reduction are given in Figure 5.7.
0.5
Figure 5.7 Change in exponent b with reduction ratio
3 4 5 6 7 8 9 1
-0.1
b = -0.190 + 0.0127RR + 13.4RR(0.0998)
RR
Reduction ratio
0.0
0.1
0.2
0.3
0.4
b = -0.501 + 0.501RR

E
x
p
o
n
e
n
t
,

b
0 1 2
-0.5
-0.4
-0.3
-0.2
0

Substituting the functions that relate the coefficient a and the exponent b to the
reduction ratio into Equations 5.6 or 5.7 gives the following general expression relating
t
10
and K
Ic
2
/E:
| |
( )
( )
( )
0.501 0.501
2
10
0.190 0.0127 13.4 0.0998
2
10 3.40
7.01 7.01 for 1 <1.5
'
100
100 for 1.5
'
1
3.97
RR
RR
Ic
RR
Ic
K
t RR RR
E
K
t RR
E
RR
+
+ +
| |
= +
|
\ .
(
(
| |
(
=
|
(
(

[5.8]
where, RR is the reduction ratio. Equation 5.8 is applicable only over the range of
fracture toughness values determined in this study. What occurs outside of this range, at
K
Ic
2
/E values approaching zero or values greater than 80 J/m
2
, is hard to assess based on
| |
\ .
+
|
\ .
the given data. More testing is required in order to establish a wider ranging t
10
and
126
K
Ic
2
/E relationship, as well as to strengthen or modify the suggested one. Additionally,
HECT tests run at higher reduction ratios need to be performed in order to verify that the
nctions proposed in Figures 5.6 and 5.7 are acceptable. Despite these limitations it is
age function more t
n
values are
eeded. Using the one-parameter family of curves approach t
10
can be related to other t
n

values. Figure 4.3 g of that data was
used to develop Figures 5.8 and 5.9, whi show the relationship etween various t
n

values and t
10
tionships are also given in Figures
5.8 and
[5.9]
The series of equations given in 5.9 can be used along
complete breakage function for a rock of known fracture toughness exposed to a given
e data and a classification function in order to predict the
product size distribution produced by a jaw crusher when crushing the given rock
materia

fu
believed t
10
is dependent upon the fracture behavior of the rock as defined by either
fracture toughness or the strain energy release rate, which is related to K
Ic
. The fracture
toughness of a rock is dependent upon the presence, distribution, and geometry of flaws
within the rock, as well as the energy/stress applied to the rock. These factors also dictate
the size and distribution of new crack surfaces present after fracture or breakage and are
represented as t
n
values. Further study will enhance the inter-relationship between
fracture toughness and fractured size distribution.

Equation 5.8 defines t
10
for a given reduction ratio, fracture toughness, and elastic
modulus. In order to determine the complete break
n
4 showed the measured t
n
versus t
10
data. Curve fittin
ch s b
. The expressions describing those rela
5.9 and are:
0
7.68
10
t
40.9
2.17
0.281
1.5 10
0.800
2 1
4 10
10
25 10
75 10
100 10
0.556
0.271
t t
t t
t t
t
t t
t t
=
=
=
=
=
=
50 10
0.384
0.315
t t
t t
=
=
with Equation 5.8 to determine the
level of size reduction in a jaw crusher. The breakage function can then be used in
conjunction with feed siz
l.

127
Figure 5.9 Plot of functions relating small t
n
values and t
10

0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
6
7
8
9
10
t
50
= 0.384t
10
t
75
= 0.315t
10
t
100
= 0.271t
10
t
100
t
75
t
50
t
n

(
%
)
t
10
(%)
Figure 5.8 Plot of functions relating large t
n
values and
10
0 2 4 6 8 10 12 14 16 18 20
0
10
20
30
40
50
60
70
80
90
100
t
1.5
= 40.94(t
10
)
0.281
t
2
= 7.68(t
10
)
0.800
t
4
= 2.174t
10
t
10
= t
10
t
25
= 0.556t
10
t
10
t
25
t
4
t
2
t
1.5
t
n

(
%
)
t (%)
128

5.2 EXPERIMENTAL VERIFICATION OF MODELS

5.2.1 LABORATORY VERIFICATION OF E
C
AND BREAKAGE FUNCTION MODELS

It was concluded that a brief assessment of the HECT derived E
c
and breakage function
models needed to be performed before laboratory jaw crushing was conducted. This
allows for an investigation of the strength of the models strictly from a laboratory, or
baseline, perspective before scale-up variables can have an effect on the predicted results
(i.e, feed rate, jaw crusher geometry). Samples of dolomitic limestone where tested
because it had been used in the development of the Edge Notched Disk fracture
toughness test and a K
Ic
value was available. The fracture toughness of the dolomitic
limestone is 1.397 MPa m and it has an elastic modulus of 39.7 GPa.

Ten samples of the dolomitic limestone were tested in the High Energy Crushing Test
system. Except for the closed side set, which was 35.2 mm, the conditions of the HECT
system were exactly the same as used for the six Luck Stone rocks. The 10 samples were
disk specimens with an average diameter of 50.22 mm and thickness of 24.82 mm. The
average reduction ratio was 1.43. The average specific comminution energy of the ten
samples was determined and the pooled fragments sieved for breakage function analysis.
The specific comminution energy results are shown in table 5.2 along with the predicted
value of E
c
as given by Equation 5.3. The measured t
10
value and the t
10
value predicted
by Equation 5.9 are also in Table 5.2. The actual breakage function and the predicted
breakage function given by Equation 5.9 are shown in Figure 5.10.

Table 5.2 Actual and predicted E
c
and t
10
for dolomitic limestone
Parameter Measured Predicted
E
c
0.295 kWh/t 0.305 kWh/t
t
10
6.0 % 6.6 %

Table 5.2 indicates that the E
c
model developed from the experimental program is
fairly accurate for the given reduction ratio of 1.43. Since this reduction is within the
bounds of the measured data it is not surprising the actual and predicted E
c
values are in
good agreement. It remains to be seen if the model holds at higher reduction ratios,
which could not be tested here due to the limited availability of rocks and the re-
configuration of the HECT system required for small closed side sets and intense
secondary crushing.

The measured and predicted breakage functions also show fairly good agreement,
particularly at the larger progeny sizes. As the daughter size decreases past 0.1 (which
represents t
10
) t ge distribution
increases. This is because the relationships between t
10
and the smaller t
n
values (t
50, 75,
100
)
he deviation between the measured and predicted breaka
given in Equation 5.9 are not as strong as those between t
10
and the larger t
n
values.
129
The average coefficient of determination for the relationships between t
10
and t
n
s coarser
than t
10
is 0.85, and between t
10
and the finer t
n
s it is 0.60.
1
Measured
Predicted

5.2.2 PREDICTIO L

A l
ure toughness based model for the prediction of capacity. The crushing
tests were conducted on rock samples from Boscobel, Culpeper, Shadwell, and
Spotsylvania in accordance with the procedure outlined in section 3.5. Appendix IV
contains data collected from the laboratory crushing trials.

Verification of breakage function (product size) model
The closed side set and open side set of the crusher were 7.94 mm and 14.29 mm
respectively. The throw of the crusher was 6.35 mm and the crusher speed was
approximately 300 rpm. Based on the feed size distributions shown in Figure 3.15 the
Figure 5.10 Measured and predicted breakage function for dolomitic limestone
0.01 0.1 1
0.01
0.1
B

(
d
i
,
d
1
)
d
i
/d
1
N OF PRODUCT SIZE USING A ABORATORY JAW CRUSHER
aboratory scale crusher was used to test the ability of the breakage function model,
when used with the feed size distribution and a classification function, to predict the
product size coming out of an actual jaw crusher. The lab-scale crusher was also used to
develop a fract
maximum reduction ratio seen by any particle in the crusher was around 4. Using
Whitens crusher model, determined from mass balance equations, and given in Equation
2.4, the product size was predicted using the breakage function model given in the
previous section along with a classification function and the feed size distribution. The
130
breakage function for each particle size in the feed was determined using Equations 5.8
and 5.9 where reduction ratio was defined as the particle size divided by the closed side
set. The classification function was calculated for each feed particle using Equation 2.5.
A MathCAD program was developed in order to perform matrix calculation of the
product size using the feed size distribution (mass fraction in each size) vector, the
diagonal classification function matrix, and the strictly lower triangular breakage function
matrix. The product size distribution vector was determined in terms of the mass fraction
retained in each size class. This vector in turn was converted to a percent passing
distribution in order to compare it to the actual, measured size distribution of the product.

The actual product size distribution from the jaw was determined by sampling and
f the cylindrical disk specimen during HECT testing. Since most of
the feed material was in the 19.1 to 25.4 mm size range, the breakage function developed
in this study using the disk specimen implies that a large portion, if not all, of the product
will pass at sizes half of 19.1 to 25.4, or 9.5 to 12.7 mm. In each case 100% of the
product was predicted to pass at 12.7 mm. If irregular particles were used during single
particle breakage analysis more agreement would be expected between the actual and
predicted distributions at the larger sizes. The
distributions are very similar in each case with the major difference between the two
curves being in the proportion of material in each size class. The Boscobel granite
product distributions showed the most deviation, with about a constant 5% difference (by
% passing) between the actual and predicted percentage passing.

are minimal. The rocks crushed in the lab jaw were also the same as ones
used in the development of the breakage function model, thus the applicability of the
odel to rocks outside the measured data range is still in question. In the same vein, the
maximum reduction seen in the lab crushing experiments was about 4, which is outside
the range used in model development but not too far (maximum HECT reduction ratio
was 3). Since jaw crushers can see higher reduction ratios up the applicability of the
breakage function to all jaw crushing reduction ratios remains to be tested. Despite these
limitations the fracture toughness based breakage function model can be used to provide
a reasonable estimate of the product size coming out of a jaw crusher given the feed size
distribution and the closed side set of the crusher.
sieve analysis. Approximately 9 kg of product material was sampled, sieved using a Ro-
Tap, and weighed for each rock tested. The actual and predicted results are shown in
Figures 5.11 through 5.14.

It can be seen from Figures 5.11 - 5.14 that the actual and predicted results are in
fairly good agreement. The predicted product size distribution deviates at the 12.7 mm
size due to the use o
shapes of the predicted and actual
Based on the results of the laboratory jaw crushing experiment, the suggested fracture
toughness based model for the determination of the breakage function seems reasonable.
However, lab crushing is a small-scale attempt at reproducing very large-scale
conditions. The feed size used may not be representative of what is seen in actual
crushing conditions and the effects of jaw plate wear, throughput, and other operating
conditions
m
131
132
Figure 5.11 Pr i vania granite edicted and actual product s ze distributions for Spotsyl
1 10 30
0
10
20
30
40
50
60
70
80
90
100
%

p
a
s
s
i
n
g
Particle size (mm)
Spotsylvania granite
Actual
Predicted
Figure 5.12 P l metabasalt redicted and actual product size distributions for Shadwel
1 10 30
0
10
20
30
40
50
60
70
80
90
100
%

p
a
s
s
i
n
g
Particle size (mm)
Metabasalt
Actual
Predicted

Figure 5.13 Predicted and actual product size distributions for Culpeper siltstone
1 10 30
0
10
20
30
Particle size (mm)
100
100
90 Actual
Predicted
Siltstone
40
50
60
70
80
%

p
a
s
s
i
n
g
Figure 5.14 Predicted and actual product size distributions for Boscobel granite
90 Actual
Predicted

1 10 30
0
10
Particle size (mm)
Boscobel granite
20
70
80
30
40
50
60
%

p
a
s
s
i
n
g
133
5.3 MODEL FOR VOLUMETRIC CAPACITY OF A JAW CRUSHER

In section 2.1.3 equations were given for the volumetric capacity of a jaw crusher
based on the dimensions of the crusher, its operational settings (closed side set and
throw), its speed, the average feed size, and an undefined parameter related to the nature
of the material. The laboratory jaw crusher trials were used to determine the actual
volumetric capacity of the jaw, for a given rock, with the expectation that the K
3

parameter could be defined in terms of fracture toughness.

For each rock the average particle size and bulk density of the feed were determined,
and the time required to crush the feed material was recorded. The volumetric capacity
for each rock was then calculated. Table 5.3 lists the measured capacities for each rock.

Table 5.3 Rock specific volumetric capacities of laboratory jaw crusher
Bulk density Mass Volume Time Vol. capacity
Rock Type
kg/m
3
kg m
3
sec m
3
/hr
Culpeper siltstone 1702 38.80 0.0228 160 0.513
Shadwell metabasalt 1837 45.97 0.0250 195 0.462
Spotsylvania granite 1691 42.97 0.0254 169 0.541
Boscobel granite 1670 47.14 0.0282 172 0.591

Equation 2.12 gives the volumetric capacity of a jaw crusher. The predicted capacity
is the ideal capacity (i.e., strictly a function of the crusher geometry and operational
set ed
ze, throw and gape, and the nature of the material, respectively. Since K
3
is undefined it
ctual capacity
easured using the lab crusher is a function of K
3
. This can be expressed as (Sastri,
tings) normalized by three parameters, K
1
, K
2
, and K
3
, which are functions of the fe
si
is assumed that the deviation between the predicted capacity and the a
m
1994):

3
actual
predicted
V
K
V
= [5.10]
where (recalling Equations 2.12 and 2.13),
( )
1 2
2
450
60 0.5
predicted
g
V Nw CSS T K K
N
| |
|

1
0.85
avg
K
|
2.5
F
.
| |
6.5
2
1.92 10
T
G
G
K

\ .
=
and, N is the speed of the crusher in rpm
w is the width of the jaws in m
CSS is the closed side set in m
T is the throw in m
= +
\
= [5.11]
134
D is the vertical depth between the jaws in m
G is the crusher gape in m
g is gravitational acceleration in m/s
2
.
Equation 5.10 was used to determine the predicted volumetric capacity for each rock,
which in turn was compared to the actual volumetric capacity using Equation 5.11. It
was anticipated that K
3
would be a function of either fracture toughness or strain energy
release rate. Figure 5.15 is plot of K
3
versus fracture toughness. It can be seen that K
3

decreases with fracture toughness, or in other words, the volumetric capacity decreases
1.0
with increasing fracture toughness. K
3
decreases from a value of one since it is expected
that the capacity of a material with little to no hardness is strictly a function of the
crusher geometry, operational settings, and feed size. Again, the Culpeper siltstone
shows unique behavior in that it has the highest fracture toughness yet its value of K
3
, and
in turn the volumetric capacity, was not the lowest among the four rocks crushed. The
upward trend is a result of the large amount of shatter occurring in the siltstone particles
upon fracture. This behavior has been discussed in detail in the previous sections. The
increased shatter occurring in the siltstone results in a uniform progeny size distribution
consisting of more finely sized fragments. In regards to volumetric capacity, these
fragments pass through the crusher without experiencing much secondary breakage and
exit the crushing chambering rather quickly. Additionally, the lack of large fragments
that result primarily from cleavage enables more feed material to enter the crushing
chamber. Based on this observation K
3
was plotte
Figure 5.15 K
3
parameter versus fracture toughness
Ic
d against the strain energy release rate,
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
0.0
0.1
0.2
0.3
0.4
0.5
0.6
K

3
1.8 2.0
0.7
0.8
0.9
K (MPa M
0.5
)
135
Figure 5.16 K
3
parameter versus strain energy release rate
0 10 20 30 40 50 60 70 80 90 100
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
K

3
G (J/m
2
)
136
which was previously related to the product size and takes into account the dominant
type of fracturing seen in a rock (i.e, cleavage, shatter). Figure 5.16 shows K
3
versus G
for the four rocks crushed.

Figure 5.16 indicates that the volumetric capacity decreases with an increase in strain
energy release rate to a limiting value (which corresponds to K
3
equal to 0.54) and then
begins to increase as the strain energy release rate increases past 50 J/m
2
. The
relationship between K
3
and G (where G equals K
Ic
2
/E) can be fit with an exponential
expression of the following form:
( )
2 2
'
3
0.0987 0.0055 0.8995 0.9674
'
Ic
K
Ic E
K
K
E
| |
= + +
|
\ .
[5.12]
The volumetric capacity of a jaw crusher can be determined by substituting Equation 5.12
into Equation 2.12.

The relationship given in 5.12 is limited by the small number of rock types (4)
crushed in the laboratory jaw crusher. Verification of the fracture toughness based K
3

and volumetric capacity models were not performed. Additional testing on the laboratory
scale, or even on a large scale using actual jaw crushers, needs to be performed in order
to assess the applicability of the given model.
CHAPTER 6. CONCLUSIONS AND RECOMMENDATIONS


6.1 CONCLUSIONS

In todays crushing environment the need for optimization of jaw crusher selection and
operation to meet the economic requirements of quarrying has become imperative.
Aggregate processing plants require rugged, massive, and expensive equipment, and
misjudgments in design are difficult and expensive to correct. The selection of a jaw
crusher for application in the aggregate industry is primarily based on technical literature
provided by crusher manufacturers, experience, and cost. Charts and graphs provide data
on electric power requirements, crusher size, as well as expected capacities for a given
material and closed side set. These charts and graphs have also been incorporated into
computer programs to aid in crusher selection. Jaw crusher selection is also heavily
influenced by the subjective judgment/experience of individuals, which results in the
conservative selection and operation of jaw crushers.

In order to account for material variations manufacturers rely on a suite of laboratory
tests. The uniaxial compression and Bond impact crushability test are two of the most
common tests used to rank materials relative to their hardness/crushability. Each of these
tests has limitations and neither adequa aterials resistance to fracture o
e breakage process that occurs in a jaw crusher. The plates of a jaw crusher are used to
tely describes a m r
th
apply compressive forces that induce tensile stresses within particles, causing fracture.
Any description of the nature of rock material, relative to jaw crushing, should be based
on, at the very least, tensile strength. Although jaw crushers are used extensively the lack
of understanding relative to their operational characteristics, as well as a reliance on an
inappropriate, single set of material properties, makes selection of a proper machine
difficult, and optimization of in-place crushers almost non-existent.

The focus of jaw crusher selection should be on the prediction of power consumption,
product size, and capacity. Each of these parameters is a function of the mechanical
aspects of the jaw crusher as well as the rock material being crushed, and prediction of
these parameters requires the proper characterization of the mechanical nature of rock in
regards to the jaw crushing environment and operational settings. This will allow for the
rapid estimation of power consumption, product size, and capacity for a given rock being
crushed under a given set of operating conditions.

Consideration of the physics of particle fracture and the mechanisms that drive
fracture initiation is necessary in the evaluation and improvement of jaw crushing
operations. Rocks broken in jaw crushers fail at stress levels well below what is
predicted based on stress-strain behavior or compressive strength, due to induced tensile
stresses (rocks are much weaker in tension than compression) and the presence of cracks.
Rocks by their nature contain inherent flaws in the form of grain boundaries, voids/pores,
be ss dding planes, foliations, etc., on both small and large scales. These cracks act as stre
137
concentrators, and the increase in stress at these locations is proportional to the geometry
f the crack.

. The importance in terms of
w crushing operations is that fracture is dependent on applied loads (i.e., energy) and
The mechanisms of fracture also control the distribution of progeny particle sizes and
specifi sics is
the foundation for describing size reduction processes. Energy-size reduction
relationships are related to Griffiths energy criterion and the presence of Griff cks.
The energy criterion states that enough potential energy must be released in order to
overcome a materials resistance to crack propagation, requiring an increase in the work
d rces a on the al. Th e amou f energy into
re a par Therefo energy ed for
of size reduction achieved, or the size distributi lting f oth
ependent upon the presence and distribution of Griffith flaws.
of rocks, and establishing a relationship between flaw geometry and fracture
strength is the most fundamental aspect of fract
defined by fracture toughness, the critical level a
terizes the crack opening displacements that
crusher. Induced tensile stresses open pre-
existing flaws and ultimately lead to crack propagation and particle failure. When
considering jaw crushing, mode I fracture toughness is the material property most
representative of the breakage process occurring due to point contact loading and it is
dependent upon the same parameters (applied stress/energy inpu
geometry) that control the power consumption and product size of jaw crushers.

In order to relate rock fracture toughness to the power consumption, product size, and
etric capacity of jaw crushers, a series of laboratory experiments, comprised of
le breakage, and small scale jaw crushing, had to
re toughness test had to be developed and verified
tive, estimation of rock fracture toughness could be achieved.
o
Griffiths theory expounds a relationship between crack size and fracture stress.
Griffith hypothesized that fracture occurs when the energy supplied by an external force,
or by the release of stored strain energy, is greater than the energy of the new crack
surface. Griffiths work defined a constant critical value that when equal to the product
of the applied stress and crack length will result in fracture
ja
the presence, distribution, and size of flaws, and both need to be accounted for when
characterizing a rock materials resistance to fracture.

c fracture mechanisms produce specific fragment sizes, and thus particle phy
ith cra
one by external fo cting materi is is th nt o input
ducing the size of ticle. re the requir fracture and the amount
on resu rom fracture, are b
d

Modifications to Griffiths theory have led to the development of the field of fracture
mechanics. Fracture mechanics provides quantitative methods for characterizing the
behavior of an intact material as it fractures due to crack growth. Fracture mechanics is
applicable to problems of rock breakage since the presence of inherent flaws and
discontinuities within geologic materials result in failure. These flaws control the
fracture
ure mechanics. This relationship is
bove which crack extension occurs.
Mode I fracture toughness properly charac
occur in particles being broken in a jaw
t, flaw distribution and
volum
fracture toughness testing, single partic
be undertaken. In addition a new fractu
so that rapid, and representa
138
A major drawback in studying size reduction processes from a fracture mechanics point
complexity of rock fracture toughness testing.

comparison. When compared to the Semi-Circular bend test, a more common and
accepted fracture toughness test, the END test yields almost identical results.

Seven rocks, provided by aggregate producer Luck Stone Corporation, were tested for
fracture toughness, specific comminution energy, and breakage function. Specific
comminution energy and breakage function were determined using a unique single
particle breakage device, the High Energy Crushing Test (HECT) system. The HECT
system is a specially configured materials testing system that is capable of simulating a
wide range of crushing operations. For this study the HECT system was configured to
deliver a sinusoidal crushing blow at a speed of 225 rpm, average jaw crushing
conditions. Since power consumption and product size are functions of the crusher
settings as well as the material being crushed, the closed side set of the HECT system
was varied, resulting in single particle breakage tests run at two reduction ratios, 1.5 and
3. Core based, disk specimens were employed in the HECT system in order to eliminate
the effect of microstructure changes due to blasting (since the END test also used rock
core). Although some size/geometry dependency was seen in the breakage function
results, the experimental program performed in this study provides a unique core based
approach that can be used to determine the jaw crushing requirements of a given rock.

The results of the fracture toughness and HECT system tests indicate a strong,
proportional relationship between fracture toughness and specific comminution energy.
Additionally, fra inution energy
more strongly th trength. For the
the road in terms of required breakage energy. It was observed during HECT breakage
of view has been the difficulty and
The Edge-Notched Disk Wedge Splitting test yields representative fracture toughness
values (in accordance with Level I testing) using a simple, core based specimen. It
requires little specimen preparation time since it employs a straight-through notch.
Testing, load application, and data collection are straightforward. The test enables a large
number of rock types and specimens to be tested allowing for quick indexing and
rocks tested in this study a linear relationship most adequately describes the relationship
between fracture toughness and specific comminution energy. As fracture toughness
increases so does the energy required to achieve a certain amount of size reduction (as
defined by the reduction ratio, the ratio of the initial particle size to the closed side set).
As the reduction ratios increases so does the specific comminution energy. Therefore,
the specific comminution energy is function of both the fracture toughness of the rock
material and the operational characteristics of the jaw crusher, as defined by the closed
side set.

Of the rocks tested only one, a siltstone, did not fall within the 95% confidence limit
for the suggested linear relationship between fracture toughness and specific
comminution energy. The siltstone had the highest fracture toughness yet was middle of
cture toughness was shown to be related to specific comm
an any other material property tested, including tensile s
139
of the siltstone that the rock tended to shatter upon fracture, producing a wide range of
progeny sizes. The lack of daughter fragments created by cleavage resulted in less
secondary crushing and consequently lower levels of specific comminution energy. This
observation was substantiated quantitatively by the uniform distribution of progeny
particles seen in the siltstones breakage function.

When the the feed size distribution, the mass flow at each size, the probability of
breakage and the idling power are taken into account, the relationship between fracture
toughness and specific comminution energy can be used to define a model for the
prediction of jaw crusher power consumption, given by:

| |
1
0.428
1
0.511 0.511 for 1 1.5
0.215 for 1.5
j
c i Ic i i n
i
j
c i Ic i i n
i
P RR K C x P
P RR K C x P RR
=
=
= + + <
(
= +

RR

where, P
c
is the power consumption of the crusher in kW
RR
i
is the reduction ratio for a particle of size I
K
Ic
is the fracture toughness of the rock being crushed in MPa m
C
i
is the probability of breakage for particle size i
x
i
is the mass flow of particle size i in metric tons per hour
P
n
is the power drawn by the crusher under no load in kW.
Since the max to be made
in regards to what occurs at higher reduction ratios, and the applicability of the power
tween fracture toughness and specific comminution energy.
he common procedure of expressing breakage functions as mixtures of separate size
pop
ck was taken into account. Additional relationships
between t
10
and various t
n
values were also evident, thus allowing for the determination
of a rocks breakage function based soley on its fracture toughness and

nergy release rate of the rock. This is because for rocks with higher
rain energy release rates shatter fracture is more prevalent and more energy is released
rogeny undergo additional breakage in
imum reduction ratio used in this study was 3, assumptions had
consumption model to higher reduction ratios remains to be tested.

The relationship between breakage function and fracture toughness was not as strong
or as evident as the one be
T
ulations (fine and coarse) was not employed due to the lack of a relationship between
the size proportion and distribution parameters and fracture toughness. Instead, the one-
parameter family curves approach was used. In this case a relationship between t
10
, the
characteristic size reduciton parameter, and fracture toughness was evident but only when
the elastic modulus, E, of the ro
elastic modulus.
The relationship demonstrated by the data shows that t
10
increases with K
Ic
2
/E,
which is the strain e
st
as new crack surface energy. As the reduction ratio is increased, the relationship between
t
10
and K
Ic
2
/E inverses, and t
10
decreases with increasing strain energy release rate. The
low strain energy release rate rocks are subjected to more secondary breakage than the
higher K
Ic
2
/E rocks because the low K
Ic
2
/E rocks initially produce large progeny
fragments as a result of cleavage. These p
140
comparison to the fragments of the high strain energy release rate rocks, which are
roduced by shatter and have more particles of varying size.
e breakage function is:
p

The model for th
| |
( )
( )
10
0.190 0.0127 13.4 0.0998
2
10 3.40
0.281
1.5 10
7.01 7.01 for 1 <1.5
'
100
100 for 1.5
'
1
3.97
40.9
RR
Ic
RR
Ic
t RR RR
E
K
t RR
E
RR
t t
+ +
= +
|
\ .
(
(
(
=
|
(
(

=
0.800
2 10
7.68
2.17
t t
t t
=
=
( ) 0.501 0.501
2
RR
K
+
| |
4 10
| |
| |
\ .
+
|
\ .
10 10
25 10
50 10
75 10
100 10
0.556
0.384
0.315
0.271
t t
t t
t t
t t
t t
=
=
=
=
=

where, t
10
is the percent passing at a progeny size 1/10
th
of the parent size
K
Ic
is the fracture toughness of the rock being crushed in MPa m
t
n
values are percent passing at a progeny size 1/n
th
of the parent size
and, E is given by,
' for conditions of plane stress E E =

2
' for conditions of plane strain
1
E
E
v
=

where, v is Possions ratio.

As is the case with the power consumption model, the applicability of the breakage
function model to high reduction ratios is based strictly on assumptions. Additionally,
since the relationship between t
10
and K
Ic
2
/E is not well defined the given breakage
function model is only recommended for rocks with strain energy release rates that fall
ithin the range measured in this study. More testing is required in order to establish a
wid
2
w
er ranging t
10
and K
Ic
/E relationship, as well as to strengthen/modify the suggested
one.

The strain energy release rate was also used in the development of the volumetric
capacity model. Although volumetric capacity was shown to decrease with increasing
fracture toughness, the siltstone once again displayed deviatoric behavior, as the ability of
the laboratory jaw crusher to handle the siltstone actually increased compared to less
141
tough rocks. When using strain energy release rate it was observed that the volumetric
capacity initially decreases with increasing K
Ic
2
/E but begins to level off and then
crease as the strain energy release rate increases beyond 50 J/m
2
. Again, this is due to
3
in Sastris volumetric capacity
quation was defined. The adjusted model for volumetric capacity of a jaw crusher is:
in
the propensity for high strain energy release rate rocks to shatter upon fracture. The
increased shatter occurring in the siltstone resulted in more finely sized daughter
fragments. These fragments passed through the crusher without incurring secondary
breakage and exited the crushing chambering rather quickly. The lack of large fragments
that result primarily from cleavage also allowed more feed material to enter the crushing
chamber.

Using K
Ic
2
/E the previously undefined parameter K
e

( )
( )
1 2 3
1 2 3 2
60 0.5 at low speeds
450
60 0.5 at high speeds
DT
V Nw CSS T K K K
G CSS T
g
V Nw CSS T K K K
N
| |
= +
|
|
+
\ .
| |
= +

|
\ .

where, N is the speed of the crusher in rpm
w is the width of the jaws in m
CSS is the closed side set in m
T is the throw in m
D is the vertical depth between the jaws in m
( )
G is the crusher gape in m
and, K
1
, K
2
, K
3
are given by,

( )
2
2.5
1
6.5
2
2
'
3
0.85
1.92 10
0.0987 0.0055 0.8995 0.9674
'
Ic
avg
T
G
K
Ic E
F
K
G
K
K
K
E

| |
=
|
\ .
=
| |
|
\ .

where, F
avg
is the average feed size in m.

Full verification of the models for power consumption and volumetric capacity were
not performed. Verification of the model strictly for specific comminution energy was
performed, along with verification of the breakage function model using the HECT
system. The breakage functon model was tested further using the laboratory jaw
= + +
crusher,
y comparing the actual product size distribution with the predicted product size
dis
b
tribution based on Whitens crusher model and the breakage function.

For the HECT system verification experiment, dolomitic limestone, which had
previously been tested for fracture toughness using the END test, was used. The fracture
toughness based models for specific comminution energy and t
10
predicted that the
142
dolomitic limestone would require 0.305 kWh per metric ton of energy to crush, with
brekage resulting in a t
10
value of 6.6%. The actual values were 0.295 kWh/ton and
.0%, respectively. The entire predicted breakage function also showed good agreement
agreement
ith the actual product size produced by the crusher. There is some deviation at larger
on
quirements. This would result in lower capital cost and lower operating costs. It is
p
iven.

6
with the measured function, with some deviation occurring at large t
n
values since the
breakage function model loses power with increasing t
n
.

The prediction of product size for the laboratory scale jaw crusher was performed
using four rock types. Employing Whitens crusher model, based on a mass balance
approach, the product size was predicted using the breakage function, classification
function, and feed size distribution. The predicted product size showed good
w
sizes due to the effect of the disk specimen used to develop the breakage function model.
In two cases the predicted percent passing was slightly higher than measured, and in the
other two instances it was slightly lower. Overall the predicted product size distribution
was within 5%, in terms of percent passing (by weight) a given particle size.

Unfortunately full scale verification, using jaw crushers in operation at Luck Stone
sites, could not be performed. One reason is because the feed size distribution entering
Luck Stones jaw crushers has never been determined. Secondly, Luck Stone employs
essentially the same size jaw crusher at each of its operations and comparison of the
predicted power consumption to the installed electric drive would be baseless since the
same drive is used at each operation. But at the same time this is the argument for using
a new approach to selecting jaw crushers. Based on the model suggested here, the
toughest rock tested would consume 230% more power than the weakest for the same
feed size and average reduction ratio. In turn, a smaller crusher, driven by a smaller
motor, could be used for the weaker rock as long as it is able to meet producti
re
im ortant to realize that this would also require a shift away from avoiding secondary
breakage at all costs, the main reason large crushers are employed almost universally. Or
it would require a change in blast specifications such that the feed material entering the
jaw crusher will not bridge.

For cases such as Luck Stones, where jaw crushers are already in place, the breakage
function and volumetric capacity models developed in this study can be used to
determine feed size distributions and operational settings (i.e., closed side set) that
optimize the product size and production capacity. This would improve downstream
processes since control over their feed sizes would be excercised. Blasting would also be
improved since guidelines for muck size disitribution, which becomes the feed material,
could be g
This study clearly demonstrates that a fracture mechanics approach to the study of
size reduction processes is appropriate, and most thoroughly characterizes the process of
rock fragmentation in crushing equipment. Rock fracture toughness is related to the
amount of energy required to reduce a particle to a given size. It can also be related to
143
the resultant fractured size distribution and the volumetric capacity of a jaw crusher.
Equipment selection and optimization can now be performed using a material property
that properly represents the fracture process occurring in a jaw crusher. Fracture
toughness can be rapidly determined using the END test, and along with feed size data,
and the average amount of size reduction, K
Ic
can be used to determine the power
onsumption, product size, and the volumetric capacity of a jaw crusher processing a
giv
6.2 RECOMMENDATIONS

All size reduction processes are dependent upon the mechanical parame
reduction and the nature of the rock being broken. Therefore all size reduction and rock
fragmentation processes, beginning with blasting, should be analyzed using fracture
ed using some measure of its
in energy release rate.
test was verified in this study,
size and notch radius.
ticularly in regards to
veloped in this study
eed to either verified or modified, based on additional testing, for reduction ratios
e
ower consumption of cone crushers require scale-up factors, and the same may be
c
en rock. The process can also be reversed so that fracture toughness, along with
product size and/or desired capacity, can be used to optimize the feed size or closed side
set of the crusher.

ters of the size
mechanics, with the nature of the rock being characteriz
ability to resist fracture, either fracture toughness or stra

Although the applicability of the Edge Notched Disk
there is some evidence that the results are dependent on grain
Additional refinement of the END test needs to be performed, par
using a chevron notch instead of a straight through notch.

The models for power consumption and the breakage function de
n
greater than 3. The breakage function model also requires more testing in order to further
clarify the relationship between fragmented product size and fracture toughness/strain
energy release rate. Testing at higher reduction ratios will also improve the relationships
between t
10
and other t
n
vales used to determine the breakage size distribution. Since
only 6 rock types were used in this study the models proposed could be
strengthened/adjusted simply by testing more rocks that are both inside and outside of the
measured range. Specifically, more sedimentary rocks types should be tested in order to
determine the how representative of sedimentary rocks the Culpeper siltstone is.

All the models proposed here require full-scale verification. Similar models for th
p
expected for the jaw crusher power consumption model. The volumetric capacity model
is based only on the results of lab-scale crushing of four rocks, and it too may be subject
to a scale-up factor since actual jaw crushers are on the order of 10 times bigger than the
lab jaw used in this study. The feed size distribution used in the lab crushing experiments
was fairly narrow and the scale-up effect of feed size and distribution must also be
accounted for.

144
This investigation relied solely on the testing of rock core specimens (except for the
laboratory jaw tests). In reality, rocks fed to a jaw crusher have undergone a change in
eir microstructure due to blasting. Although it would be expected that similar
rela hness, specific comminution energy, and
breakage function, a similar study should be undertaken using rock specimens prepared
from blasted rock. The effect of blasting on the distribution and size of flaws, and thus
fracture toughness, can be examined.

Likewise, single particle breakage testing should be done on irregular particles that
more closely resemble those found in the feed. The influence of the disk specimen
geometry on the breakage distribution can then be determined, as well as it ability to
mimic an irregular particle under point contact loading.

It is clear that fracture mechanics and fracture toughness can be used to model and
predict the behavior of both rock and machine when considering jaw crushing
applications. Jaw crushers can now be selected or optimized based on only a few
parameters as long as the fracture toughness of the rock being broken is known. Proper
characterization of the rock, along with the exclusion of overemphasized factors such as
design experience and secondary breakage, allows for the selection of suitably sized
equipment and/or the optimization of the product and production capacity. This results in
reduced capital costs, reduced operating costs, efficient production, and an overall
improvement in the entire size reduction process.

ort in studying
eds to be carried out in
pplication of
ther size reduction applications.


th
tionships hold between fracture toug
It is also apparent that this study is just the beginning, and that more eff
size reduction from a rock fracture mechanics point of view ne
order to refine and verify the findings outlined here, as well as to widen the a
fracture mechanics to o
145
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156








APPENDIX I











END RESULTS: FRACTURE TOUGHNESS
157
D t a F
v
F
w
K
Ic
29
BG-W-2 47.447 25.095 21.920 367 367 0.657
BG-W-3 47.473 26.289 21.920 416 416 0.709
BG-W-5 47.498 25.502 22.047 289 289 0.511
23 0.042
9 0.195
BG-W-10 47.447 24.841 22.123 294 293 0.536
AVG 0.601
STDEV 0.083
Did not fracture at notch
Table A1.1 Fracture toughness results for Boscobel granite
mm mm mm N N
MPa m
1/2
BG-W-1 47.498 25.375 21.920 73 73 0.1
Specimen
BG-W-4 47.473 24.511 21.920 48 48 0.088
BG-W-6 47.498 24.790 22.123 23
BG-W-7 47.473 25.603 21.971 131 131 0.229
BG-W-8 47.523 26.416 22.123 344 344 0.589
BG-W-9 47.498 25.121 21.996 109 10
158
Requirement t
Specimen
a D D - a Size of FPZ
mm mm mm mm mm mm
BG-W-3 18.871 OK OK OK OK 0.301
0.156
BG-W-6 0.067 OK OK OK OK 0.001
BG-W-8 13.027 OK OK OK OK 0.207
023
BG-W-10 10.772 OK OK OK OK 0.172
Table A1.2 Specimen size requirements for Boscobel granite
BG-W-1 0.628 OK OK OK OK 0.010
BG-W-2 16.157 OK OK OK OK 0.257
BG-W-4 0.289 OK OK OK OK 0.005
BG-W-5 9.792 OK OK OK OK
BG-W-7 1.974 OK OK OK OK 0.031
BG-W-9 1.426 OK OK OK OK 0.
159
D t a F
v
F
w
K
Ic
2
CG-W-3 47.574 25.527 20.599 862 861 1.404
CG-W-4 47.574 25.044 20.701 805 804 1.343
CG-W-5 47.625 25.248 20.701 1028 1027 1.699
CG-W-6 47.574 25.070 20.599 1044 1043 1.732
CG-W-7 47.650 25.298 20.726 1016 1014 1.676
CG-W-8 47.625 24.943 20.726 762 761 1.277
CG-W-9 47.574 25.146 20.726 935 934 1.556
CG-W-10 47.574 24.232 20.650 799 798 1.375
AVG 1.454
STDEV 0.195
Table A1.3 Fracture toughness results for Culpeper gray siltstone
mm mm mm N N
MPa m
1/
Specimen
CG-W-1 47.600 25.121 20.676 749 748 1.243
CG-W-2 47.600 24.790 20.574 736 735 1.231
160
Requirement t D D - a Size of FPZ
mm mm mm mm mm mm
CG-W-1 8.035 OK OK OK 1.023
CG-W-2 7.876 OK OK OK OK 1.003
CG-W-3 1.304
CG-W-4 9.380 OK OK OK OK 1.194
CG-W-5 15.008 OK OK OK OK 1.911
CG-W-6 15.587 OK OK OK OK 1.985
CG-W-7 14.602 OK OK OK OK 1.859
CG-W-8 8.474 OK OK OK OK 1.079
CG-W-9 12.582 OK OK OK OK 1.602
CG-W-10 9.824 OK OK OK OK 1.251
Specimen
Table A1.4 Specimen size requirements for Culpeper gray siltstone
a
OK
10.242 OK OK OK OK
161
162

D t a F
v
F
w
K
Ic
mm mm mm N N
MPa m
1/2
CR-W-1 47.625 24.130 22.250 863 862 1.623
CR-W-2 47.600 25.451 22.200 686 685 1.221
CR-W-3 47.600 28.550 22.885 792 791 1.307
CR-W-4 47.574 24.917 22.250 958 957 1.748
CR-W-5 47.549 25.578 22.225 1043 1041 1.853
CR-W-6 47.523 24.460 22.276 823 822 1.535
CR-W-7 47.625 24.943 22.225 921 919 1.673
CR-W-8 47.625 25.451 22.250 558 557 0.995
CR-W-9 47.625 23.063 22.225 734 733 1.442
CR-W-10 47.549 25.121 22.098 897 895 1.611
AVG 1.501
STDEV 0.262
Specimen
Table A1.5 Fracture toughness results for Culpeper red siltstone
163

Requirement t a D D - a
Size of FPZ
mm mm mm mm mm mm
CR-W-1 14.373 OK OK OK OK 0.610
CR-W-2 8.131 OK OK OK OK 0.345
CR-W-3 9.318 OK OK OK OK 0.395
CR-W-4 16.679 OK OK OK OK 0.708
CR-W-5 18.726 OK OK OK OK 0.795
CR-W-6 12.853 OK OK OK OK 0.546
CR-W-7 15.271 OK OK OK OK 0.648
CR-W-8 5.398 OK OK OK OK 0.229
CR-W-9 11.349 OK OK OK OK 0.482
CR-W-10 14.156 OK OK OK OK 0.601
Specimen
Table A1.6 Specimen size requirements for Culpeper red siltstone
164

D t a F
v
F
w
K
Ic
mm mm mm N N
MPa m
1/2
LD-W-1 47.600 26.060 21.768 588 636 1.081
LD-W-2 47.574 25.527 21.768 650 704 1.222
LD-W-3 47.574 25.908 21.742 754 816 1.395
LD-W-4 47.625 24.841 21.742 682 738 1.313
LD-W-5 47.625 25.730 21.793 667 722 1.243
LD-W-6 47.600 25.959 21.742 753 814 1.387
LD-W-7 47.574 24.968 21.742 604 654 1.159
LD-W-8 47.625 26.441 21.819 679 734 1.233
LD-W-9 47.574 26.238 21.768 724 784 1.324
LD-W-10 47.600 26.086 21.742 647 700 1.186
AVG 1.254
STDEV 0.101
Specimen
Table A1.7 Fracture toughness results for Leesburg Diabase
Requirement t a D D - a Size of FPZ
mm mm mm mm mm mm
LD-W-1 9.435 OK OK OK OK 0.400
LD-W-2 12.048 OK OK OK OK 0.511
LD-W-3 15.696 OK OK OK OK 0.666
LD-W-4 13.917 OK OK OK OK 0.591
LD-W-5 12.473 OK OK OK OK 0.529
LD-W-6 15.533 OK OK OK OK 0.659
LD-W-7 10.835 OK OK OK OK 0.460
LD-W-8 12.259 OK OK OK OK 0.520
LD-W-9 14.141 OK OK OK OK 0.600
LD-W-10 11.359 OK OK OK OK 0.482
Specimen
Table A1.8 Specimen size requirements for Leesburg diabase
165

D t a F
v
F
w
K
Ic
mm mm mm N N
MPa m
1/2
SMB-W-1 47.447 24.486 21.920 710 709 1.300
SMB-W-2 47.447 24.409 21.946 750 749 1.380
SMB-W-3 47.447 25.197 21.844 1221 1219 2.163
SMB-W-4 47.447 24.105 21.920 1274 1272 2.370
SMB-W-5 47.447 24.917 21.996 729 727 1.317
SMB-W-6 47.396 24.232 21.946 576 575 1.069
SMB-W-7 47.346 24.892 21.946 930 928 1.683
SMB-W-8 47.422 24.917 21.996 812 811 1.469
SMB-W-9 47.422 24.816 21.971 914 912 1.657
SMB-W-10 47.396 24.994 21.971 983 982 1.772
AVG 1.411
STDEV 0.215
Specimen
Failed to meet specimen size reqiurements
Table A1.9 Fracture toughness results for Shadwell metabasalt
166

Requirement t a D D - a Size of FPZ
mm mm mm mm mm mm
SMB-W-1 12.166 OK OK OK OK 0.376
SMB-W-2 13.701 OK OK OK OK 0.424
SMB-W-3 33.672 NO NO OK NO 1.041
SMB-W-4 40.425 NO NO OK NO 1.250
SMB-W-5 12.481 OK OK OK OK 0.386
SMB-W-6 8.220 OK OK OK OK 0.254
SMB-W-7 20.396 OK OK OK OK 0.631
SMB-W-8 15.539 OK OK OK OK 0.480
SMB-W-9 19.767 OK OK OK OK 0.611
SMB-W-10 22.601 OK NO OK OK 0.699
Specimen
Table A1.10 Specimen size requirements for Shadwell metabasalt
167

D t a F
v
F
w
K
Ic
mm mm mm N N
MPa m
1/2
SG-W-1 47.473 25.806 21.971 554 553 0.965
SG-W-2 47.447 25.375 21.971 485 484 0.859
SG-W-3 47.447 25.273 21.895 608 608 1.078
SG-W-4 47.473 25.095 22.022 531 530 0.954
SG-W-5 47.498 24.740 21.971 400 400 0.726
SG-W-6 47.473 24.765 22.022 279 279 0.508
SG-W-7 47.371 25.248 21.920 392 392 0.699
SG-W-8 47.498 25.222 22.123 541 540 0.971
SG-W-9 47.473 25.222 21.946 479 478 0.851
SG-W-10 47.523 25.222 22.047 460 459 0.821
AVG 0.843
STDEV 0.165
Specimen
Table A1.11 Fracture toughness results for Spotsylvania granite
168
169

Requirement t a D D - a
Size of FPZ
mm mm mm mm mm mm
SG-W-1 14.533 OK OK OK OK 0.617
SG-W-2 11.525 OK OK OK OK 0.489
SG-W-3 18.144 OK OK OK OK 0.770
SG-W-4 14.194 OK OK OK OK 0.602
SG-W-5 8.235 OK OK OK OK 0.350
SG-W-6 4.032 OK OK OK OK 0.171
SG-W-7 7.621 OK OK OK OK 0.323
SG-W-8 14.714 OK OK OK OK 0.624
SG-W-9 11.314 OK OK OK OK 0.480
SG-W-10 10.528 OK OK OK OK 0.447
Specimen
Table A1.12 Specimen size requirements for Spotsylvania granite

D t a F
v
F
w
K
Ic
mm mm mm N N
MPa m
1/2
TG-W-1 47.193 26.162 21.285 636 619 1.035
TG-W-2 47.142 25.883 21.311 654 636 1.079
TG-W-3 47.168 25.756 21.234 829 807 1.368
TG-W-4 47.168 25.654 21.234 666 648 1.103
TG-W-5 47.168 25.806 21.234 813 791 1.339
TG-W-6 47.219 24.917 21.184 589 573 1.000
TG-W-7 47.092 24.460 21.184 592 576 1.029
TG-W-8 47.142 25.603 21.234 609 593 1.012
TG-W-9 47.168 25.959 21.260 657 640 1.077
TG-W-10 47.142 25.679 21.209 622 605 1.028
AVG 1.045
STDEV 0.037
Specimen
Failed to meet specimen size reqiurements
Table A1.13 Fracture toughness results for Thornberg granite
170
Requirement t a D D - a
Size of FPZ
mm mm mm mm mm mm
TG-W-1 14.998 OK OK OK OK 0.637
TG-W-2 16.296 OK OK OK OK 0.692
TG-W-3 26.194 NO NO OK NO 1.112
TG-W-4 17.052 OK OK OK OK 0.724
TG-W-5 25.093 OK NO OK OK 1.065
TG-W-6 13.991 OK OK OK OK 0.594
TG-W-7 14.827 OK OK OK OK 0.629
TG-W-8 14.345 OK OK OK OK 0.609
TG-W-9 16.257 OK OK OK OK 0.690
TG-W-10 14.807 OK OK OK OK 0.628
Specimen
Table A1.14 Specimen size requirements for Thornburg granite
171


















APPENDIX II

HECT RESULTS: SPECIFIC COMMINUTION ENERGY
172
D t Mass F E T P
mm mm g kN pts MPa kWh/t
BG-E-1-1 47.473 25.654 115.3 10.898 42.665 5.697 0.103
BG-E-1-2 47.473 25.603 116.3 14.887 67.242 7.798 0.161
BG-E-1-3 47.473 25.984 123.4 35.993 8.889 18.576 0.020
BG-E-1-4 47.422 25.984 123.4 41.936 87.393 21.666 0.197
BG-E-1-5 47.473 25.705 115.3 18.975 39.197 9.899 0.094
BG-E-1-6 47.498 26.035 123.4 43.693 7.809 22.493 0.018
BG-E-1-7 47.473 25.832 117.4 24.909 0.463 12.931 0.001
BG-E-1-8 47.549 25.552 117.4 19.073 3.205 9.994 0.008
BG-E-1-9 47.447 25.806 122.4 51.281 70.323 26.662 0.160
BG-E-1-10 47.574 25.324 113.3 15.764 46.524 8.330 0.114
BG-E-1-11 47.473 25.502 119.4 17.516 59.564 9.211 0.139
BG-E-1-12 47.549 25.070 111.3 13.233 18.058 7.067 0.045
BG-E-1-13 47.549 25.781 117.4 17.125 59.472 8.893 0.141
BG-E-1-14 47.473 25.171 115.3 24.326 82.403 12.960 0.198
BG-E-1-15 47.498 25.730 113.3 12.067 24.360 6.286 0.060
BG-E-1-16 47.473 25.654 114.3 21.604 85.755 11.293 0.208
BG-E-1-17 47.498 25.375 115.3 21.697 61.704 11.461 0.149
BG-E-1-18 47.498 25.806 118.4 19.269 46.797 10.008 0.110
BG-E-1-19 47.498 25.756 113.3 16.738 32.339 8.710 0.079
BG-E-1-20 47.473 26.035 114.3 15.666 38.749 8.069 0.094
AVG 11.900 0.105
STDEV 5.840 0.065
1 pts = 1 watt sec
FREQ = 3.8 Hz
SPAN = 284
THROW = 50.80 mm
SET PT = 410
CSS = 31.75 mm
P = Mass specific crushing energy
Table AII.1 HECT results for Boscobel granite at reduction ratio 1
M = Specimen mass
F = Peak crushing force
E = Total crushing energy
T = Tensile strength
Specimen
D = Specimen diameter
t = Specimen thickness
173

D t W F E T P
mm mm g kN pts MPa kWh/t
BG-E-2-1 47.523 25.654 110.3 11.676 42.032 6.097 0.106
BG-E-2-2 47.295 26.543 115.3 14.207 88.610 7.205 0.213
BG-E-2-3 47.371 26.441 113.3 9.634 13.547 4.897 0.033
BG-E-2-4 47.320 25.756 111.3 17.418 92.112 9.098 0.230
BG-E-2-5 47.447 26.162 120.4 23.379 155.383 11.990 0.359
BG-E-2-6 47.498 26.187 120.4 24.615 133.220 12.598 0.307
BG-E-2-7 47.473 25.705 116.3 15.470 63.534 8.071 0.152
BG-E-2-8 47.422 26.060 117.4 12.748 135.602 6.567 0.321
BG-E-2-9 47.523 25.502 115.3 12.748 57.332 6.696 0.138
BG-E-2-10 47.523 26.111 119.4 23.641 39.300 12.129 0.091
BG-E-2-11 47.473 25.705 117.4 20.238 75.158 10.558 0.178
BG-E-2-12 47.523 26.416 120.4 17.027 145.752 8.635 0.336
BG-E-2-13 47.523 25.781 119.4 34.957 72.519 18.164 0.169
BG-E-2-14 47.523 25.908 118.4 25.594 72.192 13.233 0.169
BG-E-2-15 47.523 26.162 120.4 27.151 135.790 13.902 0.313
BG-E-2-16 47.498 26.543 120.4 25.496 196.590 12.874 0.454
BG-E-2-17 47.447 26.416 121.4 21.697 165.771 11.021 0.379
BG-E-2-18 47.498 25.933 119.4 23.739 91.050 12.269 0.212
BG-E-2-19 47.473 26.314 120.4 19.073 122.132 9.720 0.282
BG-E-2-20 47.396 25.527 113.3 9.243 60.464 4.863 0.148
AVG 10.029 0.230
STDEV 3.456 0.110
1 pts = 1 watt sec
FREQ = 3.8 Hz
SPAN = 224
THROW = 50.80 mm
SET PT = 406
CSS = 16.00 mm
P = Mass specific crushing energy
Table AII.2 HECT results for Boscobel granite at reduction ratio 2
M = Specimen mass
F = Peak crushing force
E = Total crushing energy
T = Tensile strength
Specimen
D = Specimen diameter
t = Specimen thickness
174
D t W F E T P
mm mm g kN pts MPa kWh/t
CG-E-1-1 47.625 26.822 129.8 65.875 146.230 32.830 0.313
CG-E-1-2 47.727 26.492 128.8 52.740 NA 26.555 NA
CG-E-1-3 47.574 26.721 128.8 57.117 157.110 28.604 0.339
CG-E-1-4 47.600 25.832 122.7 50.792 93.486 26.298 0.212
CG-E-1-5 47.600 26.111 123.7 70.741 174.160 36.234 0.391
CG-E-1-6 47.650 26.695 127.8 53.349 136.560 26.700 0.297
CG-E-1-7 47.600 26.949 128.8 46.028 147.640 22.843 0.318
CG-E-1-8 47.625 26.695 127.8 53.518 153.410 26.799 0.334
CG-E-1-9 47.600 26.518 127.8 53.714 104.140 27.091 0.226
CG-E-1-10 47.625 26.695 129.8 47.389 137.093 23.729 0.293
CG-E-1-11 47.625 26.899 130.8 65.292 42.735 32.447 0.091
CG-E-1-12 47.625 26.314 126.8 57.802 52.096 29.362 0.114
CG-E-1-13 47.625 26.848 130.8 52.157 80.812 25.969 0.172
CG-E-1-14 47.625 27.229 131.8 53.714 236.690 26.370 0.499
CG-E-1-15 47.625 26.594 129.8 55.560 NA 27.927 NA
CG-E-1-16 47.600 26.924 128.8 52.059 219.860 25.860 0.474
CG-E-1-17 47.650 26.721 128.8 55.075 198.612 27.537 0.428
CG-E-1-18 47.650 26.721 129.8 54.879 123.510 27.439 0.264
CG-E-1-19 47.625 26.137 126.8 50.787 182.050 25.975 0.399
CG-E-1-20 47.600 26.111 125.8 46.219 41.576 23.674 0.092
AVG 27.512 0.292
STDEV 3.219 0.123
1 pts = 1 watt sec
FREQ = 3.8 Hz
SPAN = 284
THROW = 50.80 mm
SET PT = 410
CSS = 31.75 mm
P = Mass specific crushing energy
Specimen
D = Specimen diameter
t = Specimen thickness
M = Specimen mass
F = Peak crushing force
E = Total crushing energy
T = Tensile strength
Table AII.3 HECT results for Culpeper gray siltstone at reduction ratio 1
175

D t W F E T P
mm mm g kN pts MPa kWh/t
CG-E-2-1 47.650 24.892 121.4 45.152 177.472 24.234 0.406
CG-E-2-2 47.600 25.933 126.4 45.152 105.540 23.286 0.232
CG-E-2-3 47.650 25.425 122.4 45.632 205.113 23.978 0.466
CG-E-2-4 47.650 25.578 124.4 65.479 146.622 34.202 0.327
CG-E-2-5 47.650 25.705 124.4 46.415 170.931 24.124 0.382
CG-E-2-6 47.600 24.587 119.4 50.009 196.981 27.203 0.458
CG-E-2-7 47.625 25.730 124.4 53.323 188.210 27.702 0.420
CG-E-2-8 47.600 25.730 125.4 46.708 140.070 24.279 0.310
CG-E-2-9 47.625 25.197 122.4 49.235 244.950 26.120 0.556
CG-E-2-10 47.600 24.511 118.4 47.776 131.061 26.069 0.308
CG-E-2-11 47.625 25.552 123.4 51.183 102.832 26.776 0.232
CG-E-2-12 47.574 25.375 123.4 50.890 116.780 26.837 0.263
CG-E-2-13 47.600 26.162 227.050 26.016 0.499
CG-E-2-14 47.600 25.781 125.4 43.497 142.160 22.565 0.315
CG-E-2-15 25.375 123.4 .554 158.700 25.578 0.357
CG-E-2-16 0.552
CG-E-2-17 47.625 25.400 123.4 52.936 46.233 27.859 0.104
CG-E-2-18 47.650 27.559 132.4 60.230 153.372 29.199 0.322
CG-E-2-19 47.625 26.060 126.4 61.200 139.650 31.392 0.307
CG-E-2-20 47.650 25.171 122.4 44.275 81.963 23.500 0.186
AVG 26.286 0.350
STDEV 2.854 0.119
1 pts = 1 watt sec
FREQ = 3.8 Hz
SPAN = 224
THROW = 50.80 mm
SET PT = 406
CSS = 16.00 mm
P = Mass specific crushing energy
Specimen
D = Specimen diameter
t = Specimen thickness
M = Specimen mass
F = Peak crushing force
E = Total crushing energy
T = Tensile strength
Table AII.4 HECT results for Culpeper gray siltstone at reduction ratio 2
126.4 50.890
47.625 48
47.600 25.781 125.4 47.825 249.290 24.810
176
177

D t W F E T P
mm mm g kN pts MPa kWh/t
CR-E-1-1 47.574 26.848 131.4 55.707 183.590 27.766 0.388
CR-E-1-2 47.600 25.908 125.4 55.756 142.040 28.783 0.315
CR-E-1-3 47.625 26.289 127.4 61.204 147.412 31.121 0.321
CR-E-1-4 47.625 26.187 127.4 60.146 90.996 30.701 0.198
CR-E-1-5 47.625 26.238 127.4 71.911 92.529 36.636 0.202
CR-E-1-6 47.650 26.721 128.4 56.632 33.229 28.316 0.072
CR-E-1-7 47.549 27.051 131.4 54.395 92.853 26.922 0.196
CR-E-1-8 47.650 26.035 125.4 52.446 34.285 26.914 0.076
CR-E-1-9 47.650 26.670 128.4 45.828 72.848 22.957 0.158
CR-E-1-10 47.650 26.848 130.4 59.843 56.112 29.780 0.120
CR-E-1-11 47.650 24.740 121.4 55.164 154.000 29.790 0.352
CR-E-1-12 47.625 26.949 130.4 70.149 68.577 34.795 0.146
CR-E-1-13 47.676 25.908 126.4 52.446 142.793 27.031 0.314
CR-E-1-14 47.574 27.305 130.4 57.215 302.560 28.040 0.645
CR-E-1-15 47.650 25.883 125.4 48.070 192.811 24.813 0.427
CR-E-1-16 47.701 25.298 123.4 66.751 201.000 35.214 0.453
CR-E-1-17 47.600 26.619 130.4 48.848 172.720 24.543 0.368
CR-E-1-18 47.650 25.730 125.4 58.869 NA 30.567 NA
CR-E-1-19 47.650 26.822 130.4 58.994 240.730 29.385 0.513
CR-E-1-20 47.650 26.035 125.4 46.219 180.122 23.718 0.399
AVG 28.890 0.298
STDEV 3.691 0.156
1 pts = 1 watt sec
FREQ = 3.8 Hz
SPAN = 284
THROW = 50.80 mm
SET PT = 410
CSS = 31.75 mm
P = Mass specific crushing energy
Specimen
D = Specimen diameter
t = Specimen thickness
M = Specimen mass
F = Peak crushing force
E = Total crushing energy
T = Tensile strength
Table AII.5 HECT results for Culpeper red siltstone at reduction ratio 1

D t W F E T P
mm mm g kN pts MPa kWh/t
CR-E-2-1 47.600 25.629 123.4 63.055 61.943 32.906 0.139
CR-E-2-2 47.600 25.171 122.4 57.606 65.419 30.608 0.149
CR-E-2-3 47.523 26.416 127.4 52.157 169.870 26.450 0.370
CR-E-2-4 47.600 26.162 126.4 24.424 158.400 12.486 0.348
CR-E-2-5 47.574 25.654 124.4 54.782 19.440 28.575 0.043
CR-E-2-6 47.625 25.832 125.4 62.374 NA 32.277 NA
CR-E-2-7 47.676 25.959 128.4 38.924 142.820 20.023 0.309
CR-E-2-8 47.574 26.340 127.4 63.344 214.040 32.181 0.467
CR-E-2-9 47.600 25.121 121.4 56.534 43.105 30.099 0.099
CR-E-2-10 47.625 25.629 126.4 46.121 189.552 24.056 0.417
CR-E-2-11 47.701 25.527 125.4 49.044 214.000 25.641 0.474
CR-E-2-12 47.625 24.994 123.4 51.570 88.018 27.581 0.198
CR-E-2-13 47.549 25.883 126.4 47.287 305.770 24.461 0.672
CR-E-2-14 47.574 25.400 122.4 46.415 195.310 24.453 0.443
CR-E-2-15 47.625 25.679 126.4 43.884 NA 22.844 NA
CR-E-2-16 47.625 25.883 128.4 55.756 76.973 28.796 0.167
CR-E-2-17 47.523 26.340 127.4 51.179 317.941 26.029 0.693
CR-E-2-18 47.650 25.095 124.4 46.415 34.331 24.710 0.077
CR-E-2-19 47.574 26.060 125.4 61.787 174.622 31.727 0.387
CR-E-2-20 47.650 26.035 127.4 43.497 144.390 22.321 0.315
AVG 26.411 0.320
STDEV 4.932 0.192
1 pts = 1 watt sec
FREQ = 3.8 Hz
SPAN = 224
THROW = 50.80 mm
SET PT = 406
CSS = 16.00 mm
P = Mass specific crushing energy
Specimen
D = Specimen diameter
t = Specimen thickness
M = Specimen mass
F = Peak crushing force
E = Total crushing energy
T = Tensile strength
Table AII.6 HECT results for Culpeper red siltstone at reduction ratio 2
178
179

D t W F E T P
mm mm g kN pts MPa kWh/t
LD-E-1-1 47.752 25.781 138.4 50.987 173.160 26.366 0.348
LD-E-1-2 47.803 27.229 145.4 48.750 220.864 23.844 0.422
LD-E-1-3 47.549 27.203 145.4 51.668 170.274 25.430 0.325
LD-E-1-4 47.574 26.975 144.4 57.290 238.110 28.420 0.458
LD-E-1-5 47.574 27.381 145.4 46.219 150.700 22.588 0.288
LD-E-1-6 47.574 26.594 140.4 53.518 183.330 26.930 0.363
LD-E-1-7 47.549 26.746 142.4 61.302 300.290 30.687 0.586
LD-E-1-8 47.625 27.000 144.4 59.794 NA 29.603 NA
LD-E-1-9 47.625 26.568 141.4 59.794 NA 30.084 NA
LD-E-1-10 47.625 26.899 143.4 45.272 NA 22.498 NA
LD-E-1-11 47.600 26.873 142.4 50.111 118.630 24.940 0.231
LD-E-1-12 47.574 27.381 146.4 49.920 213.943 24.397 0.406
LD-E-1-13 47.650 26.949 144.4 59.648 190.881 29.570 0.367
LD-E-1-14 47.600 27.280 146.4 62.957 341.770 30.866 0.648
LD-E-1-15 47.523 26.772 143.4 47.776 15.401 23.906 0.030
LD-E-1-16 47.549 27.127 144.4 68.308 343.380 33.714 0.660
LD-E-1-17 47.625 26.975 145.4 53.420 332.040 26.472 0.634
LD-E-1-18 47.600 26.975 144.4 55.271 122.160 27.404 0.235
LD-E-1-19 47.600 27.229 145.4 58.189 233.731 28.582 0.446
LD-E-1-20 47.625 27.178 144.4 45.734 14.111 22.494 0.027
LD-E-1-21 47.193 23.089 107.3 26.274 93.826 15.351 0.243
LD-E-1-22 47.523 22.454 121.4 46.317 141.230 27.633 0.323
AVG 26.445 0.371
STDEV 3.929 0.182
1 pts = 1 watt sec
FREQ = 3.8 Hz
SPAN = 284
THROW = 50.80 mm
SET PT = 410
CSS = 31.75 mm
P = Mass specific crushing energy
Table AII.7 HECT results for Leesburg diabase at reduction ratio 1
M = Specimen mass
F = Peak crushing force
E = Total crushing energy
T = Tensile strength
Specimen
D = Specimen diameter
t = Specimen thickness

D t W F E T P
mm mm g kN pts MPa kWh/t
LD-E-2-1 47.600 25.527 137.4 48.946 67.404 25.644 0.136
LD-E-2-2 47.625 26.137 139.4 46.820 NA 23.946 NA
LD-E-2-3 47.625 26.035 140.4 NA NA NA NA
LD-E-2-4 47.549 25.959 138.4 48.167 219.020 24.843 0.440
LD-E-2-5 47.625 26.391 141.4 52.082 NA 26.380 NA
LD-E-2-6 47.600 26.340 140.4 37.826 NA 19.207 NA
LD-E-2-7 47.574 26.187 139.4 47.193 290.021 24.116 0.578
LD-E-2-8 47.625 26.035 138.4 49.528 304.533 25.430 0.611
LD-E-2-9 47.625 26.162 139.4 65.973 326.283 33.708 0.650
LD-E-2-10 47.549 26.238 140.4 55.075 240.430 28.104 0.476
LD-E-2-11 47.625 26.365 141.4 63.735 135.390 32.314 0.266
LD-E-2-12 47.600 26.264 140.4 52.740 211.564 26.857 0.419
LD-E-2-13 47.625 25.857 138.4 50.013 251.110 25.855 0.504
LD-E-2-14 47.650 26.060 139.4 47.487 76.479 24.345 0.152
LD-E-2-15 47.574 26.137 139.4 47.585 253.400 24.363 0.505
LD-E-2-16 47.676 25.375 135.4 48.265 203.254 25.399 0.417
LD-E-2-17 47.625 26.518 141.4 53.714 299.070 27.077 0.587
LD-E-2-18 47.625 25.603 136.4 40.188 236.360 20.982 0.481
LD-E-2-19 47.625 25.857 137.4 47.678 344.290 24.648 0.696
LD-E-2-20 47.549 25.705 137.4 49.044 355.170 25.545 0.718
AVG 25.724 0.477
STDEV 3.275 0.174
1 pts = 1 watt sec
FREQ = 3.8 Hz
SPAN = 224
THROW = 50.80 mm
SET PT = 406
CSS = 16.00 mm
P = Mass specific crushing energy
Table AII.8 HECT results for Leesburg diabase at reduction ratio 2
M = Specimen mass
F = Peak crushing force
E = Total crushing energy
T = Tensile strength
Specimen
D = Specimen diameter
t = Specimen thickness
180

D t W F E T P
mm mm g kN pts MPa kWh/t
SMB-E-1-1 47.447 26.289 132.4 52.059 217.710 26.570 0.457
SMB-E-1-2 47.473 25.629 122.4 46.900 96.723 24.540 0.220
SMB-E-1-3 47.549 25.883 124.4 44.652 201.453 23.098 0.450
SMB-E-1-4 47.574 25.705 121.4 48.542 198.755 25.270 0.455
SMB-E-1-5 47.422 26.010 131.4 46.121 200.700 23.805 0.424
SMB-E-1-6 47.371 26.314 131.4 33.389 275.846 17.052 0.583
SMB-E-1-7 47.473 25.756 124.4 53.445 226.807 27.827 0.507
SMB-E-1-8 47.498 26.086 132.4 45.948 231.300 23.608 0.485
SMB-E-1-9 47.523 25.832 124.4 42.224 215.783 21.897 0.482
SMB-E-1-10 47.447 26.060 131.4 45.152 187.250 23.247 0.396
SMB-E-1-11 47.498 25.578 121.4 44.641 145.765 23.392 0.334
SMB-E-1-12 47.523 25.781 122.4 45.126 187.956 23.448 0.427
SMB-E-1-13 47.473 25.654 120.4 40.068 200.047 20.945 0.462
SMB-E-1-14 47.549 25.425 120.4 30.358 120.210 15.986 0.277
SMB-E-1-15 47.473 25.984 123.4 41.353 129.331 21.342 0.291
SMB-E-1-16 47.473 26.035 132.4 46.117 216.691 23.754 0.455
SMB-E-1-17 47.422 26.111 131.4 45.632 196.410 23.461 0.415
SMB-E-1-18 47.473 25.781 121.4 24.037 54.397 12.503 0.125
SMB-E-1-19 47.473 26.162 124.4 43.688 112.980 22.394 0.252
SMB-E-1-20 47.396 26.010 131.4 51.962 225.980 26.834 0.478
SMB-E-1-21 47.498 25.171 127.4 33.280 161.384 17.721 0.352
SMB-E-1-22 47.473 26.035 132.4 58.674 271.380 30.222 0.569
SMB-E-1-23 47.473 26.010 132.4 54.003 223.073 27.843 0.468
AVG 22.816 0.407
STDEV 4.102 0.112
1 pts = 1 watt sec
FREQ = 3.8 Hz
SPAN = 284
THROW = 50.80 mm
SET PT = 410
CSS = 31.75 mm
P = Mass specific crushing energy
Specimen
D = Specimen diameter
t = Specimen thickness
M = Specimen mass
F = Peak crushing force
E = Total crushing energy
T = Tensile strength
Table AII.9 HECT results for Shadwell metabasalt at reduction ratio 1
181

D t W F E T P
mm mm g kN pts MPa kWh/t
SMB-E-2-1 47.498 26.238 124.4 32.301 157.010 16.500 0.351
SMB-E-2-2 47.549 25.933 125.4 22.378 108.620 11.553 0.241
SMB-E-2-3 47.422 25.883 131.4 51.468 381.332 26.695 0.806
SMB-E-2-4 47.523 25.781 124.4 45.343 411.740 23.560 0.920
SMB-E-2-5 47.447 26.264 133.4 43.977 178.293 22.467 0.371
SMB-E-2-6 47.447 25.908 124.4 49.235 243.370 25.498 0.544
SMB-E-2-7 47.447 26.060 125.4 47.967 174.430 24.696 0.386
SMB-E-2-8 47.523 26.314 131.4 46.806 77.250 23.828 0.163
SMB-E-2-9 47.574 26.060 129.4 29.388 143.030 15.090 0.307
SMB-E-2-10 47.447 25.908 123.4 54.879 320.174 28.421 0.721
SMB-E-2-11 47.473 26.162 127.4 79.695 690.190 40.850 1.505
SMB-E-2-12 47.523 25.959 126.4 57.606 147.130 29.727 0.323
SMB-E-2-13 47.447 26.213 133.4 48.065 129.262 24.603 0.269
SMB-E-2-14 47.473 26.086 132.4 63.242 329.730 32.511 0.692
SMB-E-2-15 47.473 26.264 132.4 43.693 280.570 22.310 0.589
SMB-E-2-16 47.473 26.111 132.4 47.967 449.760 24.635 0.944
SMB-E-2-17 47.447 26.238 126.4 72.102 331.470 36.871 0.729
SMB-E-2-18 47.498 26.162 128.4 37.559 187.190 19.242 0.405
SMB-E-2-19 47.473 26.365 133.4 41.936 216.980 21.330 0.452
SMB-E-2-20 47.473 26.264 131.4 50.111 138.500 25.587 0.293
AVG 24.799 0.550
STDEV 6.885 0.323
1 pts = 1 watt sec
FREQ = 3.8 Hz
SPAN = 224
THROW = 50.80 mm
SET PT = 406
CSS = 16.00 mm
P = Mass specific crushing energy
Specimen
D = Specimen diameter
t = Specimen thickness
M = Specimen mass
F = Peak crushing force
E = Total crushing energy
T = Tensile strength
Table AII.10 HECT results for Shadwell metabasalt at reduction ratio 2
182

D t W F E T P
mm mm g kN pts MPa kWh/t
SG-E-1-1 47.447 26.848 131.4 31.430 167.524 15.707 0.354
SG-E-1-2 47.549 27.127 133.4 25.785 137.333 12.726 0.286
SG-E-1-3 47.473 26.670 127.4 8.562 NA 4.305 NA
SG-E-1-4 47.422 27.178 127.4 42.514 159.812 21.000 0.348
SG-E-1-5 47.422 26.492 124.4 23.641 150.570 11.980 0.336
SG-E-1-6 47.422 26.543 124.4 33.858 118.760 17.124 0.265
SG-E-1-7 47.447 24.028 114.3 24.726 96.719 13.807 0.235
SG-E-1-8 47.473 27.483 135.4 NA NA NA NA
SG-E-1-9 47.473 27.051 132.4 27.827 227.321 13.795 0.477
SG-E-1-10 47.422 26.899 133.4 18.099 95.680 9.033 0.199
SG-E-1-11 47.422 25.933 120.4 25.687 33.569 13.297 0.077
SG-E-1-12 47.447 27.229 133.4 22.769 130.452 11.220 0.272
SG-E-1-13 47.447 27.203 131.4 35.900 31.810 17.707 0.067
SG-E-1-14 47.473 27.076 132.4 34.441 93.259 17.058 0.196
SG-E-1-15 47.473 26.822 127.4 12.744 30.655 6.371 0.067
SG-E-1-16 47.498 26.797 130.4 26.755 62.013 13.382 0.132
SG-E-1-17 47.447 25.375 124.4 27.827 113.524 14.714 0.254
SG-E-1-18 47.447 27.076 133.4 35.513 118.510 17.598 0.247
SG-E-1-19 47.422 27.000 126.4 21.017 85.165 10.450 0.187
SG-E-1-20 47.447 26.619 123.4 33.271 89.760 16.770 0.202
AVG 13.581 0.233
STDEV 4.126 0.108
1 pts = 1 watt sec
FREQ = 3.8 Hz
SPAN = 284
THROW = 50.80 mm
SET PT = 410
CSS = 31.75 mm
P = Mass specific crushing energy
Specimen
D = Specimen diameter
t = Specimen thickness
M = Specimen mass
F = Peak crushing force
E = Total crushing energy
T = Tensile strength
Table AII.11 HECT results for Spotsylvania granite at reduction ratio 1
183

D t W F E T P
mm mm g kN pts MPa kWh/t
SG-E-2-1 47.422 26.340 122.4 33.596 160.440 17.123 0.364
SG-E-2-2 47.473 27.051 126.4 45.539 191.473 22.575 0.421
SG-E-2-3 47.498 26.086 129.4 26.657 169.610 13.697 0.364
SG-E-2-4 47.396 25.781 119.4 43.813 219.350 22.826 0.510
SG-E-2-5 47.447 26.391 130.4 24.037 148.560 12.221 0.316
SG-E-2-6 47.396 25.629 121.4 33.765 183.120 17.696 0.419
SG-E-2-7 47.422 26.187 128.4 28.316 184.861 14.516 0.400
SG-E-2-8 47.371 25.654 119.4 31.043 195.880 16.262 0.456
SG-E-2-9 47.396 25.883 122.4 25.105 87.930 13.028 0.200
SG-E-2-10 47.422 26.492 132.4 43.786 218.544 22.188 0.459
SG-E-2-11 47.447 25.984 124.4 29.188 122.543 15.072 0.274
SG-E-2-12 47.447 26.162 130.4 23.161 208.560 11.878 0.444
SG-E-2-13 47.396 25.248 125.4 21.893 167.030 11.647 0.370
SG-E-2-14 47.422 26.213 121.4 34.058 74.480 17.443 0.170
SG-E-2-15 47.422 25.832 121.4 25.687 126.260 13.349 0.289
SG-E-2-16 47.473 26.619 129.4 29.775 244.413 15.000 0.525
SG-E-2-17 47.473 25.629 124.4 20.238 33.833 10.590 0.076
AVG 15.712 0.356
STDEV 3.855 0.123
1 pts = 1 watt sec
FREQ = 3.8 Hz
SPAN = 224
THROW = 50.80 mm
SET PT = 406
CSS = 16.00 mm
P = Mass specific crushing energy
Specimen
D = Specimen diameter
t = Specimen thickness
M = Specimen mass
F = Peak crushing force
E = Total crushing energy
T = Tensile strength
Table AII.12 HECT results for Spotsylvania granite at reduction ratio 2
184

D t W F E T P
mm mm g kN pts MPa kWh/t
TG-E-1-1 47.168 27.051 127.4 43.742 76.249 21.825 0.166
TG-E-1-2 47.244 26.746 125.4 32.986 111.243 16.619 0.246
TG-E-1-3 47.168 26.873 126.4 45.734 NA 22.970 NA
TG-E-1-4 47.193 26.695 124.4 24.010 NA 12.133 NA
TG-E-1-5 47.168 27.102 126.4 34.543 134.870 17.203 0.296
TG-E-1-6 47.193 27.229 128.4 46.806 89.711 23.189 0.194
TG-E-1-7 47.168 26.568 122.4 34.445 NA 17.498 NA
TG-E-1-8 47.066 27.153 127.4 29.290 150.820 14.591 0.329
TG-E-1-9 47.168 26.568 123.4 31.136 215.930 15.817 0.486
TG-E-1-10 47.142 26.416 123.4 21.213 77.514 10.844 0.175
TG-E-1-11 47.142 27.153 126.4 33.280 148.490 16.552 0.326
TG-E-1-12 47.219 26.949 126.4 27.733 76.644 13.875 0.168
TG-E-1-13 47.193 27.254 128.4 31.136 72.996 15.411 0.158
TG-E-1-14 47.219 26.949 125.4 43.203 109.430 21.614 0.242
TG-E-1-15 47.193 22.962 107.3 45.343 270.472 26.638 0.700
TG-E-1-16 47.142 27.229 127.4 48.265 192.031 23.937 0.419
TG-E-1-17 47.168 27.229 128.4 17.321 30.161 8.586 0.065
TG-E-1-18 47.168 26.899 126.4 34.156 106.990 17.139 0.235
TG-E-1-19 47.168 26.772 125.4 40.966 125.354 20.653 0.278
TG-E-1-20 47.193 26.772 126.4 29.481 65.774 14.855 0.145
AVG 17.597 0.272
STDEV 4.718 0.152
1 pts = 1 watt sec
FREQ = 3.8 Hz
SPAN = 284
THROW = 50.80 mm
SET PT = 410
CSS = 31.75 mm
P = Mass specific crushing energy
Specimen
D = Specimen diameter
t = Specimen thickness
M = Specimen mass
F = Peak crushing force
E = Total crushing energy
T = Tensile strength
Table AII.13 HECT results for Thornburg granite at reduction ratio 1
185

D t W F E T P
mm mm g kN pts MPa kWh/t
TG-E-2-1 47.193 26.187 121.4 23.730 177.611 12.224 0.407
TG-E-2-2 47.142 26.289 122.4 23.521 222.123 12.082 0.504
TG-E-2-3 47.168 26.416 121.4 42.425 365.982 21.677 0.838
TG-E-2-4 47.168 26.264 118.4 36.976 NA 19.002 NA
TG-E-2-5 47.193 25.984 121.4 34.058 195.280 17.681 0.447
TG-E-2-6 47.168 26.010 118.4 32.502 126.730 16.866 0.297
TG-E-2-7 47.168 26.010 119.4 31.234 92.050 16.208 0.214
TG-E-2-8 47.219 25.756 117.4 29.290 108.204 15.333 0.256
TG-E-2-9 47.142 26.035 120.4 36.100 194.480 18.725 0.449
TG-E-2-10 47.193 26.365 121.4 32.399 41.661 16.577 0.095
TG-E-2-11 47.142 25.883 118.4 28.707 147.312 14.978 0.346
TG-E-2-12 47.193 26.010 120.4 24.130 NA 12.515 NA
TG-E-2-13 47.193 25.781 118.4 22.089 44.743 11.558 0.105
TG-E-2-14 47.269 26.416 121.4 26.759 103.990 13.643 0.238
TG-E-2-15 47.219 25.883 119.4 35.322 174.910 18.399 0.407
TG-E-2-16 47.219 25.705 117.4 26.857 144.524 14.087 0.342
TG-E-2-17 47.219 25.908 120.4 33.280 278.404 17.319 0.643
TG-E-2-18 47.193 26.035 120.4 36.002 139.790 18.654 0.323
TG-E-2-19 47.219 26.289 121.4 39.507 326.520 20.261 0.747
TG-E-2-20 47.219 26.060 118.4 29.775 109.230 15.404 0.256
AVG 16.160 0.384
STDEV 2.878 0.201
1 pts = 1 watt sec
FREQ = 3.8 Hz
SPAN = 224
THROW = 50.80 mm
SET PT = 406
CSS = 16.00 mm
P = Mass specific crushing energy
Specimen
D = Specimen diameter
t = Specimen thickness
M = Specimen mass
F = Peak crushing force
E = Total crushing energy
T = Tensile strength
Table AII.14 HECT results for Thornburg granite at reduction ratio 2
186


















APPENDIX III

HECT RESULTS: BREAKAGE SIZE DISTRIBUTION
187

Size Sieve Weight
Sieve + Sample
Weight
Weight Retained
Fraction
Passing
mm g g g
47.492 0.00 1.0000
38.100 0.00 1.0000
25.400 651.0 1713.5 1062.5 0.5466
19.100 665.0 1154.5 489.5 0.3377
12.700 632.5 1009.5 377.0 0.1769
6.350 615.5 810.5 195.0 0.0937
4.750 543.5 589.5 46.0 0.0740
3.360 506.5 545.0 38.5 0.0576
2.360 489.5 515.0 25.5 0.0467
1.700 475.0 494.5 19.5 0.0384
1.190 427.0 442.5 15.5 0.0318
0.850 433.5 446.5 13.0 0.0262
0.595 449.0 459.0 10.0 0.0220
0.425 434.5 442.5 8.0 0.0186
0.300 426.0 432.0 6.0 0.0160
0.210 450.0 456.5 6.5 0.0132
0.150 409.5 418.0 8.5 0.0096
0.105 384.5 386.0 1.5 0.0090
0.075 391.0 398.5 7.5 0.0058
pan 366.0 379.5 13.5 0.0000
Table AIII.1 Breakage distribution results for Boscobel granite at RR1
188

Size Sieve Weight
Sieve + Sample
Weight
Weight Retained
Fraction
Passing
mm g g g
47.464 0.00 1.0000
38.100 0.00 1.0000
25.400 651.0 1026.0 375.0 0.8401
19.100 665.0 903.0 238.0 0.7385
12.700 632.5 1252.5 620.0 0.4741
6.350 615.5 1209.5 594.0 0.2207
4.750 543.5 662.0 118.5 0.1702
3.360 506.5 612.0 105.5 0.1252
2.360 489.5 547.5 58.0 0.1004
1.700 475.0 521.0 46.0 0.0808
1.190 427.0 463.0 36.0 0.0655
0.850 433.5 462.5 29.0 0.0531
0.595 449.0 471.5 22.5 0.0435
0.425 434.5 453.0 18.5 0.0356
0.300 426.0 439.5 13.5 0.0299
0.210 450.0 462.5 12.5 0.0245
0.150 409.5 429.5 20.0 0.0160
0.105 384.5 388.0 3.5 0.0145
0.075 391.0 404.0 13.0 0.0090
pan 361.0 382.0 21.0 0.0000
Table AIII.2 Breakage distribution results for Boscobel granite at RR2
189

Size Sieve Weight
Sieve + Sample
Weight
Weight Retained
Fraction
Passing
mm g g g
47.624 0.00 1.0000
38.100 0.00 1.0000
25.400 650.5 1694.3 1043.8 0.5915
19.100 663.9 1077.0 413.1 0.4298
12.700 631.7 982.1 350.4 0.2927
6.350 614.3 982.0 367.7 0.1488
4.750 542.3 625.2 82.9 0.1163
3.360 505.6 585.2 79.6 0.0852
2.360 488.6 538.8 50.2 0.0655
1.700 474.4 514.4 40.1 0.0498
1.190 426.2 456.9 30.8 0.0378
0.850 432.9 458.0 25.2 0.0279
0.595 448.6 466.1 17.5 0.0211
0.425 433.6 447.5 14.0 0.0156
0.300 425.2 435.3 10.1 0.0117
0.210 449.4 457.3 7.8 0.0086
0.150 408.9 418.7 9.9 0.0048
0.105 384.0 385.4 1.4 0.0042
0.075 390.3 394.5 4.3 0.0025
pan 365.2 371.6 6.5 0.0000
Table AIII.3 Breakage distribution results for Culpeper gray siltstone at RR1
190

Size Sieve Weight
Sieve + Sample
Weight
Weight Retained
Fraction
Passing
mm g g g
47.621 0.00 1.0000
38.100 0.00 1.0000
25.400 651.0 1093.0 442.0 0.8220
19.100 665.0 594.5 0.5826
12.700 632.5 1083.0 450.5 0.4012
6.350 0.2039
4.750 3.5 634.0 90.5 0.1675
3.360 506.5 603.5 97.0 0.1284
2.360 489.5 568.5 79.0 0.0966
1.700 475.0 533.0 58.0 0.0733
1.190 427.0 474.5 47.5 0.0542
0.850 433.5 469.0 35.5 0.0399
0.595 449.0 474.5 25.5 0.0296
0.425 434.5 455.0 20.5 0.0213
0.300 426.0 441.0 15.0 0.0153
0.210 450.0 461.5 11.5 0.0107
0.150 409.5 422.0 12.5 0.0056
0.105 384.5 387.0 2.5 0.0046
0.075 391.0 397.5 6.5 0.0020
pan 365.5 370.5 5.0 0.0000
Table AIII.4 Breakage distribution results for Culpeper gray siltstone at RR2
1259.5
615.5 1105.5 490.0
54
191

Size Sieve Weight
Sieve + Sample
Weight
Weight Retained
Fraction
Passing
mm g g g
47.631 0.00 1.0000
38.100 0.00 1.0000
25.400 650.5 1710.2 1059.7 0.5845
19.100 663.9 1025.6 361.7 0.4426
12.700 631.7 908.7 277.0 0.3340
6.350 614.3 1052.1 437.8 0.1623
4.750 542.3 626.9 84.6 0.1292
3.360 505.7 593.7 88.0 0.0947
2.360 488.5 543.1 54.6 0.0733
1.700 474.2 518.1 43.9 0.0560
1.190 426.2 461.5 35.3 0.0422
0.850 432.9 460.2 27.3 0.0315
0.595 448.5 468.2 19.7 0.0238
0.425 433.7 449.8 16.1 0.0174
0.300 425.3 436.3 11.0 0.0131
0.210 449.5 457.6 8.1 0.0099
0.150 408.8 420.8 12.0 0.0052
0.105 384.0 384.5 0.6 0.0050
0.075 390.3 395.7 5.4 0.0029
pan 360.7 368.1 7.4 0.0000
Table AIII.5 Breakage distribution results for Culpeper red siltstone at RR1
192

Size Sieve Weight
Sieve + Sample
Weight
Weight Retained
Fraction
Passing
mm g g g
47.605 0.00 1.0000
38.100 0.00 1.0000
25.400 651.0 1346.5 695.5 0.7230
19.100 665.0 1183.5 518.5 0.5165
12.700 632.5 1075.0 442.5 0.3403
6.350 615.5 1062.0 446.5 0.1625
4.750 543.5 629.5 86.0 0.1282
3.360 506.5 596.5 90.0 0.0924
2.360 489.5 543.0 53.5 0.0711
1.700 475.0 518.0 43.0 0.0540
1.190 427.0 458.5 31.5 0.0414
0.850 433.5 459.0 25.5 0.0313
0.595 449.0 468.0 19.0 0.0237
0.425 434.5 448.5 14.0 0.0181
0.300 426.0 437.0 11.0 0.0137
0.210 450.0 459.0 9.0 0.0102
0.150 409.5 421.0 11.5 0.0056
0.105 384.5 386.5 2.0 0.0048
0.075 391.0 401.5 10.5 0.0006
pan 365.5 367.0 1.5 0.0000
Table AIII.6 Breakage distribution results for Culpeper red siltstone at RR2
193

Size Sieve Weight
Sieve + Sample
Weight
Weight Retained
Fraction
Passing
mm g g g
47.587 0.00 1.0000
38.100 0.00 1.0000
25.400 650.5 2135.6 1485.1 0.5213
19.100 664.0 1374.8 710.8 0.2922
12.700 631.8 984.7 352.9 0.1784
6.350 614.3 850.6 236.3 0.1023
4.750 542.4 588.9 46.5 0.0873
3.360 505.7 563.9 58.2 0.0685
2.360 488.7 523.4 34.7 0.0573
1.700 474.3 504.1 29.8 0.0477
1.190 426.3 448.1 21.8 0.0407
0.850 432.9 453.6 20.6 0.0341
0.595 448.5 463.6 15.1 0.0292
0.425 433.6 446.0 12.4 0.0252
0.300 425.2 434.9 9.7 0.0221
0.210 449.5 459.3 9.8 0.0189
0.150 408.9 419.4 10.6 0.0155
0.105 384.0 390.5 6.5 0.0134
0.075 390.3 401.1 10.8 0.0099
pan 365.2 395.9 30.7 0.0000
Table AIII.7 Breakage distribution results for Leesburg diabase at RR1
194

Size Sieve Weight
Sieve + Sample
Weight
Weight Retained
Fraction
Passing
mm g g g
47.608 0.00 1.0000
38.100 0.00 1.0000
25.400 651.0 1551.5 900.5 0.6763
19.100 665.0 1162.5 497.5 0.4974
12.700 632.5 1088.0 455.5 0.3336
6.350 615.5 1009.5 394.0 0.1920
4.750 543.5 639.5 96.0 0.1575
3.360 506.5 609.0 102.5 0.1206
2.360 489.5 550.5 61.0 0.0987
1.700 475.0 523.0 48.0 0.0814
1.190 427.0 465.5 38.5 0.0676
0.850 433.5 464.0 30.5 0.0566
0.595 449.0 473.0 24.0 0.0480
0.425 434.5 454.5 20.0 0.0408
0.300 426.0 441.0 15.0 0.0354
0.210 450.0 464.5 14.5 0.0302
0.150 409.5 429.0 19.5 0.0232
0.105 384.5 385.0 0.5 0.0230
0.075 391.0 416.5 25.5 0.0138
pan 361.5 400.0 38.5 0.0000
Table AIII.8 Breakage distribution results for Leesburg diabase at RR2
195

Size Sieve Weight
Sieve + Sample
Weight
Weight Retained
Fraction
Passing
mm g g g
47.477 0.00 1.0000
38.100 0.00 1.0000
25.400 651.0 1962.5 1311.5 0.4801
19.100 665.0 1148.0 483.0 0.2886
12.700 632.5 862.0 229.5 0.1976
6.350 615.5 854.0 238.5 0.1031
4.750 543.5 599.5 56.0 0.0809
3.360 506.5 562.0 55.5 0.0589
2.360 489.5 521.0 31.5 0.0464
1.700 475.0 501.0 26.0 0.0361
1.190 427.0 447.0 20.0 0.0281
0.850 433.5 449.0 15.5 0.0220
0.595 449.0 460.0 11.0 0.0176
0.425 434.5 443.0 8.5 0.0143
0.300 426.0 432.0 6.0 0.0119
0.210 450.0 455.5 5.5 0.0097
0.150 409.5 419.5 10.0 0.0057
0.105 384.5 388.5 4.0 0.0042
0.075 391.0 395.0 4.0 0.0026
pan 361.5 368.0 6.5 0.0000
Table AIII.9 Breakage distribution results for Shadwell metabasalt at RR1
196

Size Sieve Weight
Sieve + Sample
Weight
Weight Retained
Fraction
Passing
mm g g g
47.481 0.00 1.0000
38.100 0.00 1.0000
25.400 651.0 1202.0 551.0 0.7853
19.100 665.0 1144.5 479.5 0.5985
12.700 632.5 1189.5 557.0 0.3815
6.350 615.5 1116.5 501.0 0.1862
4.750 543.5 636.5 93.0 0.1500
3.360 506.5 602.0 95.5 0.1128
2.360 489.5 554.0 64.5 0.0877
1.700 475.0 523.0 48.0 0.0690
1.190 427.0 465.0 38.0 0.0542
0.850 433.5 462.5 29.0 0.0429
0.595 449.0 470.0 21.0 0.0347
0.425 434.5 451.5 17.0 0.0281
0.300 426.0 440.0 14.0 0.0226
0.210 450.0 468.5 18.5 0.0154
0.150 409.5 422.5 13.0 0.0103
0.105 384.5 389.0 4.5 0.0086
0.075 391.0 407.0 16.0 0.0023
pan 365.5 371.5 6.0 0.0000
Table AIII.10 Breakage distribution results for Shadwell metabasalt at RR2
197

Size Sieve Weight
Sieve + Sample
Weight
Weight Retained
Fraction
Passing
mm g g g
47.454 0.00 1.0000
38.100 0.00 1.0000
25.400 650.6 2179.0 1528.4 0.4035
19.100 663.9 1073.0 409.1 0.2438
12.700 631.6 859.0 227.4 0.1551
6.350 614.2 757.0 142.8 0.0993
4.750 542.2 579.0 36.8 0.0850
3.360 505.6 542.5 36.9 0.0706
2.360 488.9 511.0 22.1 0.0620
1.700 474.2 492.5 18.3 0.0548
1.190 426.0 440.9 14.9 0.0490
0.850 432.7 446.2 13.4 0.0438
0.595 448.4 460.9 12.5 0.0389
0.425 433.3 446.6 13.3 0.0337
0.300 425.3 439.8 14.6 0.0280
0.210 449.4 463.3 14.0 0.0226
0.150 408.8 425.2 16.4 0.0162
0.105 383.9 391.9 8.0 0.0131
0.075 390.2 401.9 11.6 0.0085
pan 365.1 386.9 21.8 0.0000
Table AIII.11 Breakage distribution results for Spotsylvania granite at RR1
198

Size Sieve Weight
Sieve + Sample
Weight
Weight Retained
Fraction
Passing
mm g g g
47.431 0.00 1.0000
38.100 0.00 1.0000
25.400 651.0 931.0 280.0 0.8686
19.100 665.0 1078.5 413.5 0.6745
12.700 632.5 1130.0 497.5 0.4410
6.350 615.5 1037.0 421.5 0.2431
4.750 543.5 626.5 83.0 0.2042
3.360 506.5 584.0 77.5 0.1678
2.360 489.5 535.5 46.0 0.1462
1.700 475.0 515.0 40.0 0.1274
1.190 427.0 459.0 32.0 0.1124
0.850 433.5 460.5 27.0 0.0997
0.595 449.0 473.5 24.5 0.0882
0.425 434.5 459.5 25.0 0.0765
0.300 426.0 450.5 24.5 0.0650
0.210 450.0 477.0 27.0 0.0523
0.150 409.5 442.5 33.0 0.0368
0.105 384.5 395.0 10.5 0.0319
0.075 391.0 417.5 26.5 0.0195
pan 365.5 407.0 41.5 0.0000
Table AIII.12 Breakage distribution results for Spotsylvania granite at RR2
199

Size Sieve Weight
Sieve + Sample
Weight
Weight Retained
Fraction
Passing
mm g g g
47.174 0.00 1.0000
38.100 0.00 1.0000
25.400 650.3 2173.4 1523.1 0.3908
19.100 663.9 948.5 284.6 0.2770
12.700 631.7 863.1 231.4 0.1845
6.350 614.3 811.1 196.8 0.1058
4.750 542.4 588.8 46.4 0.0872
3.360 505.8 547.8 42.0 0.0704
2.360 488.6 516.4 27.8 0.0593
1.700 474.3 496.5 22.2 0.0504
1.190 426.2 442.5 16.3 0.0439
0.850 433.0 446.6 13.6 0.0384
0.595 448.5 459.5 11.1 0.0340
0.425 433.6 443.5 9.9 0.0301
0.300 425.2 434.5 9.3 0.0264
0.210 449.5 459.4 9.9 0.0224
0.150 408.9 422.0 13.1 0.0171
0.105 384.0 391.5 7.5 0.0141
0.075 390.3 401.0 10.7 0.0099
pan 360.7 385.4 24.7 0.0000
Table AIII.13 Breakage distribution results for Thornburg granite at RR1
200

Size Sieve Weight
Sieve + Sample
Weight
Weight Retained
Fraction
Passing
mm g g g
47.192 0.00 1.0000
38.100 0.00 1.0000
25.400 651.0 1487.5 836.5 0.6509
19.100 665.0 979.0 314.0 0.5198
12.700 632.5 1087.5 455.0 0.3299
6.350 615.5 946.0 330.5 0.1920
4.750 543.5 625.0 81.5 0.1580
3.360 506.5 588.5 82.0 0.1237
2.360 489.5 536.0 46.5 0.1043
1.700 475.0 512.5 37.5 0.0887
1.190 427.0 457.0 30.0 0.0762
0.850 433.5 460.5 27.0 0.0649
0.595 449.0 471.0 22.0 0.0557
0.425 434.5 454.5 20.0 0.0474
0.300 426.0 444.0 18.0 0.0399
0.210 450.0 468.5 18.5 0.0321
0.150 409.5 443.0 33.5 0.0182
0.105 384.5 387.0 2.5 0.0171
0.075 391.0 407.0 16.0 0.0104
pan 361.5 386.5 25.0 0.0000
Table AIII.14 Breakage distribution results for Thornburg granite at RR2
201


















APPENDIX IV

LABORATORY JAW CRUSHING RESULTS
202

Size
mm SMB BG CR SG
35.92 1.0000 1.0000 1.0000 1.0000
25.40 0.8947 0.8030 1.0000 0.6028
19.10 0.1776 0.2833 0.5093 0.0489
12.70 0.0170 0.0385 0.0157 0.0075
4.75 0.0147 0.0090 0.0076 0.0056
0.00 0.0000 0.0000 0.0000 0.0000
Fraction passing
Table AIV.1 Feed size data for lab jaw crusher
203

Bulk
density
Mass Volume Time
Vol.
capacity
Avg. Feed
Size
K1
Predicted
Capacity
kg/m
3
kg
m
3
sec
m
3
/hr
m
m
3
/hr
Culpeper siltstone 1702 38.80 0.02280 160 0.5130 0.0157 0.8389 0.8515
Shadwell metabasalt 1837 45.97 0.02502 195 0.4618 0.0185 0.8333 0.8457
Spotsylvania granite 1691 42.97 0.02541 169 0.5413 0.0182 0.8340 0.8464
Boscobel granite 1670 47.14 0.02823 172 0.5909 0.0212 0.8266 0.8389
Closed side set (m) 0.0079
Throw (m) 0.0064
Gape (m) 0.0953
Jaw width (m) 0.1461
Jaw length (m) 0.2096
RPM 300
K2 0.7079
Rock Type
Table AIV.2 Volumetric capacity results from lab jaw crusher tests
204

Size Sieve Weight
Sieve +
Sample
Weight
Weight
Retained
Fraction
Passing
mm g g g
19.10 579.5 579.5 0.0 1.0000
12.70 552.5 910.0 357.5 0.9292
6.35 599.5 3243.5 2644.0 0.4059
4.75 533.5 1083.5 550.0 0.2971
3.36 480.5 898.5 418.0 0.2143
2.36 489.5 754.0 264.5 0.1620
pan 366.0 1184.5 818.5 0.0000
Table AIV.3 Measured product size of Boscobel granite
205

Size Sieve Weight
Sieve +
Weight
Weight
Retained
Fraction
Passing
mm g g g
19.10 1.0000
12.70 552.5 984.0 431.5 0.9211
6.35 599.5 3364.0 2764.5 0.4154
4.75 533.5 1061.0 527.5 0.3189
3.36 480.5 958.0 477.5 0.2316
2.36 489.5 779.5 290.0 0.1785
pan 366.0 1342.0 976.0 0.0000
Table AIV.4 Measured product size of Culpeper siltstone
Sample
579.5 579.5 0.0
206

Size Sieve Weight
Sieve +
Sample
Weight
Weight
Retained
Fraction
Passing
mm g g g
19.10 579.5 640.0 60.5 0.9884
12.70 552.5 1443.0 890.5 0.8184
6.35 599.5 3235.5 2636.0 0.3150
4.75 533.5 954.0 420.5 0.2347
3.36 480.5 839.0 358.5 0.1662
2.36 489.5 694.0 204.5 0.1272
pan 366.0 1032.0 666.0 0.0000
Table AIV.5 Measured product size of Shadwell metabasalt
207

Size Sieve Weight
Sieve +
Sample
Weight
Weight
Retained
Fraction
Passing
mm g g g
19.10 579.5 600.5 21.0 0.9952
12.70 552.5 926.0 373.5 0.9103
6.35 599.5 3181.5 2582.0 0.3230
4.75 533.5 883.0 349.5 0.2435
3.36 480.5 746.5 266.0 0.1830
2.36 489.5 633.5 144.0 0.1502
pan 366.0 1026.5 660.5 0.0000
Table AIV.6 Measured product size of Spotsylvania granite
208















APPENDIX V

EQUILIBRIUM ANALYSIS OF WEDGE FORCES
209
210


P
sp

P
P
v
Notch

P
( ) ( )
( ) ( )
( ) ( )
( ) ( )
Summing forces in the x and y directions:
: cos sin
2 2
: [EQ. 1]
cos sin
2 2
: sin cos
2 2
2
: [EQ. 2]
2sin 2 cos
2 2
Setting Equation 1
x sp
sp
x
v
y
v
y
F P P P
P
F P
P
F P P
P
F P

=
=

= +
=
+

( ) ( )
( ) ( )
( )
( )
( )
( )
( )
equal to Equation 2:
cos sin
2 2
2sin 2 cos
2 2
1 tan
2
2 1
1 tan
2
2tan 1 cot
2 2
v
sp
v
sp
v
sp
P
P
P
P
P
P

=
+
(

=
+

=
+

VITA

James Donovan was born on June 24, 1975 in Manalapan, NJ. He graduated from St.
John Vianney High School in 1993, where upon he enrolled at Virginia Tech. He earned
his Bachelor of Science degree in Mining Engineering in 1997. During his undergraduate
years he spent his summers working for New Hope Crushed Stone. In 1999 he earned his
Master of Science degree in Mining Engineering from Virginia Tech and completed a
thesis entitled The Effects of Backfilling on Ground Control and Recovery in Thin-Seam
Coal Mining. He stayed on at Virginia Tech as a doctoral research and teaching
assistant, and completed his PhD requirements in the summer of 2003. He currently
resides in Christiansburg, Virginia with his beautiful wife, Erin, and his glorious son,
Quinn.
211