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Bench blast modeling: Consequences of crushed zone, wave front shape, and radial cracks
AbdeI-RasouI, EIseman Ibrahim, Ph.D.
The University of Arizona, 1990
V·M·I 300 N. Zeeb Rd. Ann Arbor, MI 48106
BENCH BLAST MODELING: CONSEQUENCES OF CRUSHED ZONE,
WAVE FRONT SHAPE, AND RADIAL CRACKS
by
ELSEMAN IBRAHIM ABDEL-RASOUL
A Dissertation Submitted to the Faculty of the
DEPARTMENT OF MINING AND GEOLOGICAL ENGINEERlNG
In Partial Fulfillment of the Requirements For the Degree of
DOCTOR OF PHILOSOPHY WITH MAJOR IN MINING ENGINEERlNG
In the Graduate College
THE UNIVERSITY OF ARIZONA
1 990
1
THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE
2'
As members of the Final Examination Committee, we certify that we have read
the dissertation prepared by ELSEMAN IBRAHIM ABDEL-RASOUL
entitled BENCH BLAST MODELING: CONSEQUENCES OF CRUSHED ZONE, WAVE
FRONT SHAPE, AND RADIAL CRACKS
and recommend that it be accepted as fulfilling the dissertation requirement
for the Degree of DOCTOR OF PHILOSOPHY
C-·······~l~ 7/27/90 Date
7/27/90 Date
7/27/90 Date
7/27/90 Date
Date
Final approval and acceptance of this dissertation is contingent upon the candidate's submission of the final copy of the dissertation to the Graduate College.
I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement.
7/27/90 Dissertation Director Dr. J. Daemen Date
3
STATEMENT BY AUTHOR
This dissertation has been submitted in partial fulfilhnent of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the library.
Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgement of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.
SIGNED: '& .~ . ~
--._, .-.. -.,. .~-. ~ ...... <>---_-.- -.- ..
4
ACKNOWLEDGMENTS
The author wishes to express his sincere gratitude to his dissertation su
pervisor, Professor J. Daemen, for his support and advice provided in the course of
this study.
Acknowledgments are also due to Professors 1. Farmer, C. Glass, P. Kulati
lake, and T. Kundu, members of my dissertation committee. The author wishes to
thank Professors R. richard, J. Kemeny, and S. Harpalani for reading the manuscript
of the dissertation.
Help from Dr. Mark Borgstrom, John Saba, Linda Drew, and John Lee,
staff of the Center for Computing and Information Technology, is gratefully ac
knowledged. Also I am grateful to my colleagues Dr. Mohamed Gaballa and Dr.
Raoul Roko for their valuable discussions and assistance.
Finally, the financial support provided by the Arizona M.M.R.R.I. and the
BOM (grant number G1184104) is gratefully acknowledged.
3.7 Comparison of bending stresses, cantilever beam modelled by 10 QUAD4 elements .................... 73
3.8 Comparison of bending stresses, cantilever beam modelled by 20 Q U AD4 elements . . . . . . . . . . . . . . . . . . . . 74
Figure
3.9
3.10
3.11
3.12
3.13
3.14
3.15 3.16
3.17
3.18
3.19
3.20
3.21
3.22
4.1
4.2
4.3
4.4
LIST OF ILLUSTRATIONS--continued
Description
Comparison of bending stresses, cantilever beam modelled by 10 QUAD8 elements . . . . . . . . . . . . . . . . .
Comparison of bending stresses, cantilever beam modelled by 10 QUAD9 elements . . . . . . . . . . . . . . . . .
Mesh used to model circular hole in a plate . . . . . . .
Displacement distribution around a circular hole in a plate
Principal stress distribution around a circular hole in a plate
Comparison of tangential stresses around a circular hole in a plate . . . . . . . . . . . . . . . . . . . . . . .
Single edge crack in a plate subjected to uniaxial tension . . Mesh models single edge crack in a plate subjected to uniaxial tension. The plate modelled using QUAD4 elements ....
Mesh models single edge crack in a plate subjected to uniaxial tension. QUAD8 and QQUAD8 elements are implemented .
10
Page
· 75
· 76
· 78
· 79
•. 80
82
83
84
85
Displacement distribution through a plate containing single edge crack. The plate is modelled using QU AD4 elements (Figure 3.16) ............ .
Principal stress distribution through a plate containing single edge crack. The plate is modelled using QUAD4 elements (Figure 3.16) ............ .
Displacement distribution through a plate containing single edge crack. The plate is modelled using QQUAD8 and QUAD8 elements (Figure 3.17). . . . . . . . . . . . .
Principal stress distribution through a plate containing single edge crack. The plate is modelled using QQUAD8 and QUAD8 elements (Figure 3.17). . . . . . . . . . . .
Plots using displacement extrapolation method to calculate the stress intensity factor . . . . . . . . . . . . . . .
A model for the crushed zone around a cylindrical charge .
Effect of the velocity ratio and the characteristic impedance ratio on the scaled crushed zone diameter in granite . . .
Effect of the medium stress ratio and the detonation pressure ratio on the scaled crushed zone diameter in granite . . .
Effect of the velocity ratio and the characteristic impedance ratio on the scaled crushed zone diameter in salt . . . . .
. . 88
. .. 89
... 90
91
92
· 99
102
104
106
Figure
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
5.1
5.2
5.3
5,4
5.5
5.6
5.7
5.8
5.9
LIST OF ILLUSTRATIONS- -continued
Description
Effect of the medium stress ratio and the detonation pressure ratio on the scaled crushed zone diameter in salt . . . . .
Effect of the velocity ratio and the characteristic impedance ratio on the scaled crushed zone diameter in limestone . .
Effect of the velocity ratio and the characteristic impedance ratio on the scaled crushed zone diameter in rocks . . . .
Effect of the medium stress ratio and the detonation pressure ratio on the scaled crushed zone diameter in rocks . . . . . .
Shape of the outgoing wave front for an infinite velocity ratio
Shape of the outgoing wave front for a velocity ratio of 3 .
Shape of the outgoing wave front for a velocity ratio of 2
Shape of the outgoing wave front for a velocity ratio of 1.5
Shape of the outgoing wave front for a velocity ratio of one
Shape of the outgoing wave front for a velocity ratio of 0.8
Amplitudes of reflected distortional and dilatational waves at different angles of incidence for 1/ = 1/3 ..... .
Mesh for thick wall cylinder using nine eight noded elements
Radial and tangential stresses for thick wall cylinder using circular side elements, 2 x 2, 3 x 3 Gauss quadratures, and 9 eight noded elements . . . . . . . . . . . . . . . .
Radial and tangential stresses using 9 elements, lumped and consistent loads, 2x2 Gauss quadrature, and circular modeling of the hole boundary .......... .
Radial and tangential stresses using 9 elements, lumped and consistent loads, 2x2 Gauss quadrature, and straight line segment modeling of the hole boundary . . . . . . . .
Mesh for thick wall cylinder using 18 eight noded elements
Radial and tangential stresses using 18 eight noded elements, lumped and consistent loads, 2x2 Gauss quadrature, and circular modeling of the hole boundary . . . . . . . . . .
Radial and tangential stresses using 18 eight noded elements, lumped and consistent loads, 2x2 Gauss quadrature, and straight line segment modeling of the hole boundary . . . .
Layout of the drilling pattern showing the section modeled by the Finite Element Method . . . . . . . . . . . .
Mesh used to model the blasthole without radial cracks .
11
Page
108
110
112
114
117
119
120
121
122
124
129
140
141
143
144
145
146
148
151 153
12
LIST OF ILLUSTRATIONS--continued
Figure Description Page
5.10 Displacement field around a blasthole when radial cracks are not considered . . . . . . . . . . . . . . . . . 154
5.11 Displacement field within some selected windows when radial cracks are not considered • . . . . . . . . . . . . . . . 155
5.12 Stress field around a blast hole when radial cracks are not considered . . . . . . . . . . . . . . . . . . 160
5.13 Stress field within some selected windows when radial cracks are not considered . . . . . . . . . . . . . 161
5.14 Contour map for the scaled strain energy density around the blasthole when radial cracks are not considered .... 166
5.15 Mesh used for modeling the blasthole with radial cracks at the blasthole and at the free face ......... 170
5.16 Displacement field around a blasthole when non-pressurized radial cracks are considered . . . . . . . . . . . . . . 172
5.17 Displacement field within some selected windows when non-pressurized radial cracks are considered . . . . . . . . . 173
5.18 Stress field around a blast hole when non-pressurized radial cracks are considered . . . . . . . . . . . . . . . . . 177
5.19 Stress field within some selected windows when non-pressurized radial cracks are considered . . . . . . . . . . . . . . . 178
5.20 Contour map for the scaled strain energy density around the blasthole when non-pressurized radial cracks are considered . 183
5.21 Displacement field around a hlasthole when uniformly pressurized radial cracks are considered . . . . . . . . . . . . . . . .. 184
5.22 Displacement field within some selected windows when uniformly pressurized radial cracks are considered . . . . . . . . . 186
5.23 Stress field around a blasthole when uniformly pressurized radial cracks are considered . . . . . . . . . . . . . . 190
5.24 Stress field within some selected windows when uniformly pressurized radial cracks are considered . . . . . . . . 191
5.25 Contour map for the scaled strain energy density around the blasthole when uniformly pressurized are considered . . . . 195
5.26 Displacement field around a blasthole when linearly pressurized radial cracks are considered . . . . . . . . . . . . . . . .. 197
5.27 Displacement field within some selected windows when linearly pressurized radial cracks are considered . . . . . . . . . .. 198
Figure
5.28
5.29
5.30
5.31
5.32
5.33
5.34
5.35
LIST OF ILLUSTRATIONS- -continued
Description
Stress field around a blosthole when linearly pressurized radial cracks are considered . . . . . . . . . . • . . . .
Stress field within some selected windows when linearly pressurized radial cracks are considered . . . . . . .
Contour map for the scaled strain energy density around the blasthole when linearly pressurized radial cracks are considered . . . . . . . . . . . . . . . . . . . . .
Mesh used to model the blasthole using equivalent cavity equal to the zone of radial cracks . . . . . . . . . . . . . . .
Displacement field around a blasthole when the zone of radial cracks is replaced by an equivalent cavity . . . . . . . . .
Displacement field within some selected windows when the zone of radial cracks is replaced by an equivalent cavity . . . . .
Stress field around a blasthole when the zone of radial cracks is replaced by an equivalent cavity . . . . . . . . . . .
Stress field within some selected windows when the zone of radial cracks is replaced by an equivalent cavity . . . .
5.36 Contour map for the scaled strain energy density around the blast hole when the zone of radial cracks is
1.3
Page
203
204
208
210
212
213
217
218
replaced by an equivalent cavity . . . . . . . . . . . . . .. 222
5.37 Variation of the displacement normal to the free face with distance from the symmetry plane '. . . . . . . . . . . 226
5.38 Variation of the displacement parallel to the free face with distance from the symmetry plane . . . . . . . . . . . 227
5.39 Variation of the displacement normal to the free face with distance along the symmetry plane . . . . . . . . . . . . 228
5.40 Variation of the displacement parallel to the free face with distance along the symmetry plane within the zone of radial cracks 230
5.41 Variation of the normalized areas of the scaled strain energy density contours with tensile strength when radial cracks are not considered ',' . . . . . . . . . . . . . . . . . 233
5.42 Contour map of the scaled strain energy density when tensile strength is six times the static tensile strength when radial cracks are not considered . . . . . . . . . . . . . . . . 235
5.43 Variation of the normalized areas of the scaled strain energy density contours with tensile strength when non-pressurized radial cracks are considered . . . . . . . . . . . . . . . 236
----"--,'" .' '"---.-',."
Figure
5.44
5.45
5.46
5.47
5,48
5,49
5.50
5.51
5.52
5.53
5.54
5.55
A.l
14
LIST OF ILLUSTRATIONS- -continued
Description Page
Variation of the normalized areas of the scaled strain energy density contours with tensile strength when uniformly pressurized radial cracks are considered . . . . . . . . . . 237
Variation of the normalized areas of the scaled strain energy density contours with tensile strength when linearly pressurized radial cracks are considered . . . . . . . . . . 239
Scaled strain energy density contour map when radial cracks are not considered using the dynamic tensile strength and internal pressure 50% of the detonation pressure . . . . . . 240
Scaled strain energy density contour map when non-pressurized radial cracks are considered using the dynamic tensile strength and internal pressure 50% of the detonation pressure . . 241
Scaled strain energy density contour map when uniformly pressurized radial cracks are considered using the dynamic tensile strength and internal pressure 50% of the detonation pressure ............... 242
Scaled strain energy density contour map when linearly pressurized radial cracks are considered using the dynamic tensile strength and internal pressure 50% of the detonation pressure . . . . . . 243
Variation of the normalized areas of the scaled strain energy density with the internal pressure when non-pressurized radial cracks are considered using the dynamic tensile strength . . . . . . . . . . . . . . . 247
Contour map of the scaled strain energy density for internal pressure equal to the dynamic compressive strength when non-pressurized radial cracks are considered ....... 248
Variation of the normalized areas of the scaled strain energy density with the internal pressure when uniformly pressurized radial cracks are considered using the dynamic tensile strength . . . . . . . . . . . . . . . 249
Contour map of the scaled strain energy density for internal pressure equal to the dynamic compressive strength when uniformly pressurized radial cracks are considered . . . . 251
Variation of the normalized areas of the scaled strain energy density with the internal pressure when linearly pressurized radial cracks are considered using the dynamic tensile strength . . . . . . . . . . . . . . . 252
Contour map of the scaled strain energy density for internal pressure equal to the dynamic compressive strength when linearly pressurized radial cracks are considered 253
3.1 Shape FUnctions for the Plane Quadratic Isoparametrie Element . . 60
3.2
3.3
3.4
4.1
4.2
4.3
5.1
5.2 A.l B.l
B.2
B.3
B.4
B.5
B.6
B.7
B.8
B.9
B.lO
Summary of the Dimensions of the Apj»roximation FUnction, Compatibility, Transformation, Local Stiffness, and Global Stiffness Matrices . . . • . . . . . . . . . . . .
Summary of Tip Displacements Calculated by FEM Cantilever Beam Models . . . . . . . . . . . . . . . . . . . . . .
Percentage Error in Calculated Stress Intensity Factors . . .
Constants and Correlation Factors for the Equations Fitted to Predict the Scaled Crushed Zone Diameter in Granite
Constants and Correlation Factors for the Equations Fitted to Predict the Scaled Crushed Zone Diameter in Salt . . . . .
Constants and Correlation Factors for the Equations Fitted to Predict the Scaled Crushed Zone Diameter in Limestone
Range of the Tensile Strength Used to Calculate the Critical Strain energy Density ........... .
Range of Borehole Pressures Applied in the Models . ..
Element types used by the SABM program ...... .
Summary of Crushed Zone Data for Lithonia Granite, Data Set 1
Summary of Crushed Zone Data for Lithonia Granite, Data Set 2
Summary of Crushed Zone Data for Lithonia Granite, Data Set 3
Summary of Crushed Zone Data for Winnfield Salt
Summary of Crushed Zone Data for Marion Limestone .
Physical Properties of Lithonia Granite, Winnfield Salt, and Marion Limestone . . . . . . . . . . . .. .
Sample of Fitting Equations Obtained by the Curve Fitting Program for the Relation between the Scaled Crushed Zone Diameter and the Velocity Ratio . . . . . . . . . . . . .
Sample of Fitting Equations Obtained by the Curve Fitting Program for the Relation between the Scaled Crushed Zone Diameter and the characteristic Impedance Ratio .....
Sample of Fitting Equations Obtained by the Curve Fitting Program for the Relation between the Scaled Crushed Zone Diameter and the Medium Stress Ratio . . . . . . . . . .
Sample of Fitting Equations Obtained by the Curve Fitting Program for the Relation between the Scaled Crushed Zone Diameter and the Detonation Pressure Ratio. . .
. 66
70 . 93
103
105
109
232 245 268 291 292 293 294 295
296
297
297
298
298
16
NOTATION
Principal symbols used through the disserta.tion are summarized. If the
symbol has different meanings, it will be defined where it is used. The symbols
used here follow to a great extent the notations used by Cook (1981).
Mathematical Symbols:
[ ]
L J { } [ ]T
A rectangular or square matrix.
A row vector.
A column vector.
Matrix transpose, it applies to column and row vectors as well.
Partial differentiation with respect to the following subscript
(i.e w,:!: = 8w/8x).
Matrix inverse.
R t {alIp alIp alIp }
epresen s aat aa2 ••• aan '
where IIp is a scaler function of parameters aI, a2, ... , an.
Latin Symbols:
[ B ] Compatibility matrix (displacement strain relations).
D Displacement.
d.o.f. Degree (or degrees) of freedom.
{D} Nodal d.o.f. of a structure(global d.o.f.).
{d} Nodal d.o.f. of an element.
E Elastic modulus. .
[ E ] Matrix of elastic stiffness.
{F} Body forces per unit volume.
{f} A displacement field; {f} = {u v w} in 3-space.
Effect of charge diameter on detonation velocity. (from Dick et al, 1983, p. 14)
34
Primers should be located at the point of most confinement and/or the loca
tion of the hardest rock seam along the blasthole in order to achieve better blasting
results. Wet conditions can lower the velocity of detonation of ANFO by half (Atlas
Powder Company, pp. 205-211).
The velocity of detonation can be increased by increasing charge diameter,
density, confinement, and coupling ratio. Decreasing the particle size of the explo
sive also increases the velocity of detonation. Figure 2.8 shows the effect of charge
diameter on the detonation velocity. Strong explosives attain their maximum ve
locities at much smaller diameters than the blasting agents (Dick et al, 1983, p.
14).
The velocity of detonation of ANFO can be increased by sparingly placing a
cartridge of high von explosive every few feet in the ANFO column. This blasting
technique is referred to as alterna~e velocity. The high VOD cartridges does not
need any change in the blast design. However, field full scale bench blasts show
that it is cost-effective and improves the results of the blast. Improvements in the
blast results include: better overall fragmentation, increase in burden velocities,
increase in cast distances, and loose muck piles (Atlas Powder Company, 1987, pp.
213-226)
2.2 Shock or Stress Wave Propagation
Shock and stress wave propagation throughout the surrounding rock is the
second phase of the blasting process. It immediately follows the detonation pulse or
develops in conjunction with it. The stress waves are, in part, a result of the impact
exerted by the rapidly expanding high pressure gases. The geometry of dispersion
of these waves depends on the location of the primer, detonation velocity, and shock
wave velocity in the rock. It has been stated (Atlas Powder Company, 1987, p. 164)
that:
"In general, the stress wave propagation geometry is not dependent on the shape of the ~h.arge .... If the charge is shot, with a length to diameter ratio less than or equal to 6:1, then the disturbance is propagated in the form of an expanding sphere. If the charge is long, with a length to diameter ratio of greater than 6:1, then the disturbance is propagated in the form of an expanding cylinder. This
assumes that the detonation velocity is much greater than the rock's elastic wave velocity."
35
If the explosive charge is spherical and initiated at its center, the geometry of
the outgoing wave in the surrounding rock should be spherical as long as the rock
can be considered homogeneous, isotropic, and elastic. This spherical geometry
is not dependent on the velocity of detonation of the explosive because both the
undecomposed explosive and the rock will not sense the motion until the wave
front passes through the point. Hence, the surrounding rock is unaffected until the
detonation front arrives at the spherical interface where the wave front propagates
in the rock as a spherical surface. If the charge is cylindrical, usually the case in
real blasts, the shape of the wave front depends on the ratio of the VOD to the
elastic wave velocity of the rock. For a velocity ratio greater than one, the wave
front is sphero-conical. For a velocity ratio equal to or less than one, the wave
front is spherical. Cylindrical wave fronts exist only for an infinite velocity ratio,
which is not realistic. More discussion about the shape of the wave front around a
cylindrical charge is presented in Chapter 4.
When the outgoing compressive wave front encounters a discontinuity or
interface, some energy is transferred across the discontinuity and some is reflected
back to its point of origin. The partitioning of energy depends on the ratio of the
acoustic impedance (longitudinal wave velocity times density) of the mst medium
to that of of the second medium. For an acoustic impedance ratio less than one, the
reflected and the transmitted waves are compressive. For an acoustic impedance
ratio equal to one, all energy is transmitted to the second material, and no reflection
takes place. For an acoustic impedance greater than one, the transmitted energy is
compressive and the reflected energy is tensile. At a free face, nearly all the energy
is reflected as a tensile wave. If the burden is relatively small, most of the reflected
energy is consumed in spalling at the free face (Atlas Powder Company, 1987, pp.
165-167).
Calculations of the stress wave energy based on radial strain measurements
estimate that the wave energy forms a small fraction of the total explosive energy.
In salt, the wave energy accounts for 1.8-3.9 % of the total explosive energy (Nicolls
36
and Hooker, 1962, p. 45) and for 10-18 % in granite gneiss (Fogelson et al, 1959,
p. 15). This may imply that most of the breakage process is carried out by the gas
energy. Wilson (1987, pp. 22-25) raises a reasonable criticism for this wave energy
estimation. He states that the wave energy is underestimated for several reasons.
The wave energy has been estimated from radial strain measurements on the as
sumption that the wave propagates spherically from a pressurized cavity and that
the radial strain accounts for about 97% of the total strain energy. He questioned
these assumptions because the measurements did not account for shear waves and
because the strains caused by the wave are significant in all three principal direc
tions. This estimation of the wave energy accounts for damping and attenuation but
not for energy lost in crushing, plastic deformation, and fracturing near the charge.
This underestimates the initial stress pulse calculated by Fogelson et al. The calcu
lated wave energies were not compared to directly measured total explosive energies
but to total explosive energies based either on theoretical calculations or estimated
from ballistic morter tests. Theoretical total explosive energy calculations are based
on the assumptions of ideal and oxygen-balanced chemical reactions. The real field
detonations of commercial explosives are not ideal and hence the released explosive
energy is less than the theoretical energy. Accordingly, the measured wave energy
forms a higher fraction of the actual released explosive energy. Finally, the energy
left for blast hole gas pressurization suffers losses due to gas venting, heat flow into
the rock, and turbulent gas flow into rock fractures. Hence, wave energy may con
tribute more to the rock breakage process because it forms a higher fraction of the
total explosive energy than the above estimation.
The maximum pressure amplitude at the blasthole wall (several times the
rock compressive strength) is achieved in a very short time (a fraction of a millisec
ond). The pressure amplitude decays rapidly to a magnitude approximately equal
to the dynamic compressive strength of the rock close to the blasthole (within a few
blasthole radii). This rapid decay is due to rock pulverization, crushing, displace
ment, and gas cooling. Out of the boundary of the crushed cavity, the pressure
amplitude decay with distance is approximately exponential. As the stress wave
propagates, it creates radial and tangential (hoop) stress components. Because the
37
tensile strength of rocks is smaller than their compressive strength, a zone of radial
cracks forms outside the crushed cavity. These radial cracks can result either from
the continuation of cracks in the crushed (non-linear) zone or from newly initiated
cracks from microfractures originally existing in the rock. The radial compressive
stress, being less than the rock compre.qsive strength, is not responsible for initiating
any new fractures at this stage. When the tangential stress amplitude decays to
a certain magnitude, the wave passes through the rock causing no further crack
ing. As the stress wave propagates, expanding explosion gases penetrate and extend
the previously formed cracks and the exerted high quasi -static stresses increase the
crushed zone radius and may initiate new cracks. That is because the static strength
and the yield limit are lower than the dynamic ones. Figure 2.9 schematically illus
trates the consecutive stages in the fracture process for a fully contained explosion.
"Then a free face exists close to the blasthole, the tangential tensile stresses at the
boundary of the cavity are not uniform anymore. They are maximized at points on
the hole boundary for which 4> (Figure 2.1O(a» is a maximum:
(2.3)
Also tensile stre~ses are generated at the free face with a maximum at point
A. The radial cracks at or close to the maximum tangential stresses, propagate at
the lowest critical gas pressure and are preferred fracture directions. When they
grow toward the free f~ce, they may determine the final crater boundary in the
vicinity of the charge (Figure 2.10(b ». The reflected tensile waves at the free face
may cause spalling at the free face and may add to the rock preconditioning, which
helps gas pressure in accomplishing more rock fragmentation. It has been stated
that the wave-generated radial fractures around the crushed cavity have a diameter
of six times the hole diameter for a spherical charge and nine times the hole diameter
for a cylindrical charge (Kutter and Fairhurst, 1971).
Spalling of rock by the reflected tensile wave at a free surface has been
considered the main rock breakage mechanism by some researchers. This theory is
called the reflection theory. The role of explosion gas pressurization in the
(.)c ....... .... 0········ :
'"~ h __ .f awll, €a ..... 11.-.1_ CNIheII a_
GIew'. -' ,MI&I frlelu'"
.... '· ... "e
Figure 2.9 Consecutive stages in the fracture process of a fully contained explosion.
(after Kutter and Fairhurst, 1971)
d
Borehole \ '\.RefleCtiOn Breakage
wcusct'C(""""'><Xo::>X«WU"'1rCOfXt:'S'MCCh)(:;cK7»«< ~ 1 I 4
T ,-....---.----.. ---- -----.. , .. ,
.. , I " .... ~- I "
38
1 d-.... ........ ,...... 'l
.... I "
1 0.", Ga. Expanllon , fracture I
c.) (b)
Figure 2.10 Influence of free face surface on stress distribution around cavity.
(after Kutter and Fairhurst, 1971)
39
breakage process is considered negligible in this school of thought. According
to the reflection theory, the breakage process progresses from the free surface back
toward the shot point and the rock is pulled apart, not pushed apart. Reflection
theory has been based on cratering tests in the field (Duvall and Atchison, 1957)
and laboratory tests in which rock bars we~e fractured by detonating an explosive
charge at one end CHino, 1959, Ch. 4).
High speed photography of full scale bench blasts shows that the initial move
ment of the bench face begins after much longer elapsed time than that required
for the stress wave to travel the burden distance. Lang and Favreau (1972) used
time-distance curves for measuring burden velocity at three locations along a bench
face. The curves show elapsed times of 30, 50, and 80 msec before any burden move
ment takes place. The elapsed times include a delay time of 25 msec. This leaves 5,
25, and 35 msec before any burden movement occurs. Lang (1979) postulates that
it takes about 1 msec per foot of burden before the mass movement begins. The
stress wave needs much less travel time to arrive at the free face. This observation
not only opposes the claim that spalling is the main rock breakage mechanism but
also challenges the existence of the mechanism in full scale blasts.
A relatively new theory, called nuclei or stress wave/flaw theory, has been
formulated at the University of Maryland. This theory is based on laboratory tests
in a brittle, transparent, polyester thermosetting polymer known commercially as
Hamolite 100. Unflawed and flawed photoelastic models have been studied using
high speed cameras to capture stress wave propagation and crack formation resulting
from contained and uncontained explosions. According to the nuclei theory, stress
waves play the major role in the fragmentation process and cause a substantial
amount of crack initiation at regions remote from the blasthole. Small or large flaws,
joints, bedding planes, and other discontinuities act as nuclei for crack formation,
development or extension. The fracture network spreads with the speed of the P
and S waves. Fragmentation in blocks of rocks continues even after the blocks are
detached from the rock mass due to the trapped stress waves. Finally, the theory
claims that gas pressurization does not contribute significantly to the fragmentation
The vector of external loads applied at the structure nodes, {P}, is preferred to be
added after assembling the elements. .
Here, {r} includes many load tenns. This formulation is given only for com
pleteness. Any term which does not exist in a particular problem can be dropped.
Matrix [k] is called the element stiffness matrix and vector {r} is called element
load vector. If the body forces are eliminated from the element load vector, the
remaining tenns are the load components applied by the element to its nodes.
53
If we expand [k] and {r} to structure size, then the equations of all elements
can be assembled to form the finite element equations of the structure(Cook, 1981,
pp. 82-83). This can be represented by the equations:
numel numel
[K] = 2: [k] , {R} = {P} + 2: {r} (3.12) 1 1
and equation (3.9) becomes:
[K]{D} = {R} (3.13)
Where:
[K] = stiffness matrix of the structure;
{D} = vector of nodal displacements of the structure;
{P} = vector of loads applied at the nodes of the structure.
In case of two dimensional elasticity problems (Cook, 1981, pp. 12-15), the differ
ential operator matrix has the form:
[0] = [t ~] 8y 8%
(3.14)
In this case, the stress vector is {q} = {q % q Y q %y} and the strain vector is {E} = {E% Ey f%y}, The elasticity matrix [E] in case of plane stress has the form:
[£1= 1~P2 [~ ~ l~'] and in case of plane strain [E] has the form:
Where:
E = Young's elastic modulus;
p. = Poisson's ratio.
(3.15)
(3.16)
54
If the element coordinates are defined in a local coordinate system different from the
global coordinate system adopted for the structure, then a transformation matrix is
needed to transform the local stiffness matrix [SL] to the global stiffness matrix [SE].
This is commonly done at the element level. The following relations (llichard, 1988,
p. 13 and Cook, 1981, pp. 151-152) are used for stiffness, load, and displacement
transformations.
[SE] = [Tf[SL][T]
Where:
{J} = element displacement vector in local coordinates;
{r} = element load vector in local coordinates;
[SE] = element stiffness matrix in global coordinates;
[SL] = element stiffness matrix in local coordinates.
3.1.2 Isoparametric Formulation
(3.17)
(3.18)
In this section the isoparametric formulation will be presented(Cook, 1981,
pp. 113-142). Isoparametric elements are useful in grading a mesh from course to
fine and in modelling structures with curved edges.
Element nodes are used to define the displacements and coordinates of a
point inside the element. Symbolically, this is done using the following relations:
{u v w} = [N]{d}
. {x y z} = [N]{ c }
(3.19)
(3.20)
Vector {c} contains the global coordinates. Matrices [N] and [.IV] are the matrices
of the shape functions. They are functions of the intrinsic coordinates e , TJ, and
C. If the number of nodes is identical and if [N] and [.IV] are identical in the above
relations, the element is called isoparametric.
55
3.1.2.a The Plane Linear lsoparametric Element
Isoparametric elements are similar in formulation. The essential changes are
the addition of nodes and using different shape functions. Hence, formulation of
linear isoparametric element can be extended to more complicated elements.
The linear element is shown in Figure 3.1 (Cook, 1981, p. 117). It is an
arbitrary quadrilateral. Its natural axes e and '1 pass through the midpoints of the
opposite sides. The orientation of the e'1 coordinates is dictated by the element
assigned node numbers. The natural axes are not required to be orthogonal or to
be parallel to the x- or y-axis.
1 ~ __ ~'2 ...L
""-------------111.11
(bl
Figure 3.1 Linear quadrilateral element. (a) In the eTJ space. (b) In the xy space.
(from Cook, 1981, p. 117)
The global coordinates and displacements are defined as,
Figure 3.5 Cantilever beam models. (a) 10 QUAD4 elements. (b) 10 QUADS elements. (c) 20 QUAD4 elements. (d) 20 TRIM3 elements. Dimensions are in meters, thickness = 0.25 m.
I = moment of inertia about the principal axis (z-axis);
q z = bending stress at distance x from the free end;
Mz = bending moment at distance x from the free end;
y = distance from the principal axis.
70
The tip displacements obtained from the five FEM models are summarized in Table
3.3. The FEM displacements are expressed as ratios to the corresponding Strength
of Materials values.
Table 3.3 Summary of Tip Displacements Calculated by the FEM Cantilever Beam Models. Displacements are Expressed as Ratios to the Displacement of the Strength of Materials.
N umber of Elements Element Type ~tip Ratio
20 TRIM3 0.39
10 QUAD4 0.73
20 QUAD4 0.92
10 QUAD8 1.03
10 QUAD9 1.03
A Gauss quadrature order of 2 by 2 is used for the QUAD4 and QUAD8
elements. For QUAD9, the quadrature order of 2 by 2 gave very erroneous results.
It gave elastic displacements of upto 10+16 • Appendix A.1 includes output files for
a stick model for a cantilever beam problem. The beam tip is subjected to -6000
Newtons. The beam is modelled with two QUAD9 elements. The tip displacement
obtained by the 2x2 quadrature order is - 0.111xlO+l5 • The tip displacement
obtained by the quadrature order of 3x3 is - 0.143. These displacements are the
displacement component in the y-direction for node 15 in the output files. An order
of 3 by 3 gave good results as shown in Table 3.3 and in Appendix A.1.
71
From Table 3.3, we can see the very good results obtained by QUAD8. Its re
sults are better than those obtained by using twice the number of QUAD4 elements.
QUAD8 gave the same results as QUAD9, which needed numerical integration at
five additional Gauss points. This additional integration time is more than the total
time needed by QUAD8 (with only four integration points). The lowest accuracy
was obtained by the TRIM3 elements. However, the coefficients of their stiffness
matrix are constant and need no numerical integration. This means that more el
ements may be used to improve their results at a reasonable cost. Models using
QUAD8 and QUAD9 elements gave 3% higher deflection than the Strength of Ma
terials approach. This may be related to the boundary conditions used at the fixed
end of the beam. The use of the rollers makes the beam more flexible. In addition,
for QUAD8 elements the use of 2 by 2 quadrature order renders the beam behave
softer than reality.
Stresses are calculated at Gauss points for the quadrilateral elements. For
the triangular elements (TRIM3), they are calculated once, and are constant over
the element. For the QUAD4, QUAD8, and TRIM3 beam models, the stresses are
calculated along the lines AB and CD using the Strength of Materials formula (3.42)
to compare with the FEM calculations. Lines AB and CD are shown in Figure 3.5
(a). These lines are at 0.211 m and 0.789 m above the principal axis of the beam and
they pass through the Gauss points of the elements in the upper half of the beam.
For the QUAD9 model, stresses are calculated along three lines passing through
Gauss points. These lines are at 0.113 m, 0.5 m, and 0.887 m above the principal
axis of the beam. Bending stresses calculated using the FEM and the Strength of
Materials approach are plotted in Figure 3.6 through Figure 3.10.
Figure 3.6 shows that stresses calculated using TRIM3 elements are insensi
tive to the distance from the principal axis of the beam. The FEM bending stresses
along lines AB and CD are identical. These stresses also have the largest deviation
from the exact values. In order to improve the accuracy of the results, the number
of TRIM3 elements needs to be increased. QUAD4 shows better behaviour than
TRIM3, as shown in Figure 3.7, but there is still a large deviation from the exact
values.
e • e c c .r)
e • c c c ...
c C a
0.0 2.0 4.0 6.0 8.0 10.0
Figure 3.6
DISTANCE FROM FIXED END, METERS
Comparison of bending stresses in a cantilever beam modelled by 20 TRIM3 elements. Solid lines are stresses calculated according to strength of materials equation (3.42). Circles and triangles are FEM stresses. AB represents stresses at 0.211 m above the principal axis. CD represents stresses at 0.789 m above the principal axis (Figure 3.5 (a)). The FEM results for AB and CD are identical.
Comparison of bending stresses, cantilever beam modelled by 10 QUAD4 elements. Solid lines are stresses calculated according to strength of materials equation (3.42). Circles and triangles are FEM stresses. AB represents stresses at 0.211 m above the principal axis. CD represents stresses at 0.789 m above the principal axis (Figure 3.5 (a».
73
0 ci 0 CI CI
0 ci 0 0 Ir.I
en Z 00 E-ci A !!=O ~O z·
• en ~o en' eng ~o
~"" E-en lJ.
~o Zci ~C QO ZN ~ ~
0 • 0 0 0 ...
o ci~--------r-------~--------~--------r---~~~
0.0
Figure 3.8
2.0 4.0 8.0 8.0 10.0 DISTANCE FROM FIXED END. METERS
Comparison of bending stresses, cantilever beam modelled by 20 QUAD4 elements. Solid lines are stresses calculated according to strength of materials equation (3.42). Circles and triangles are FEM stresses. AB represents stresses at 0.211 m above the principal axis. CD represents stresses at 0.789 m above the principal axis.
74
c C c c CI
c C c c -c C;-______ .-______ .-______ .-______ ~---=~
0.0 2.0 4.0 8.0 B.O 10.0
Figure 3.9
DISTANCE FROM FIXED END, METERS
Comparison of bending stresses, cantilever beam modelled by 10 QUADS elements. Solid lines are stresses calculated according to strength of materials equation (3.42). Circles and triangles are FEM stresses. AB represents stresses at 0.211 m above the principal axis. CD represents stresses at 0.7S9 m abo"e the principal axis.
Figure 3.10 Comparison of bending stresses, cantilever beam modelled by 10 QUAD9 elements. Solid lines are stresses calculated according to streng~h of materials equation (3.42). Circles and triangles are FEM stresses. AB represents stresses at 0.113 m above the principal axis. CD represents stresses at 0.5 m a.bove the principal axis. EF represents stresses at 0.887 m above the principal axis.
76
77
Increasing the number of QUAD4 elements from 10 to 20 improved the results
significantly, (Figure 3.S). More QUAD4 elements are needed to eliminate the
deviations between the FEM results and the exact values.
QUAD8 and QUAD9 models almost produced the exact results. Ten ele
ments are used for each of these models. From Figure 3.9 and Figure 3.10, the
FEM results obtained by QUADS elements are comparable to those obtained by
QUAD9 elements. This means that QUADS should be preferred to QUAD9 for
analysis of this beam problem, because numerical integration would be carried out
at only four Gauss points compared to nine points for QUAD9 elements. At the
same time, the superiority of QUAD8 over TRIM3 and QUAD4 is very clear.
3.3.2 Circular Hole in a Plate Under Uniaxial Compression Stress Field
A circular hole of 6 m radius in a plate of of 36 m width, 48 m height, and
0.1 m thickness is modelled using QUAD4 elements. The plate is subjected to a
uniformly distributed 200 Newtons compression load in the vertical direction. Due
to symmetry, only one quarter of the plate needs to be modelled. The mesh is
shown in Figure 3.11. It consists of 33 QUAD4 elements. Figure 3.12 shows the
displacement distribution through the plate. The principal stress distribution is
illustrated in Figure 3.13. The displacements reach a maximum at the top of the
plate where they are almost directed vertically downward. They decrease and their
directions shift away from the hole as they get closer to the hole boundary. Tensile
stresses can be seen above the hole and their values decrease away from the hole.
Compressive stresses have their highest magnitudes at the side wall and they get
smaller away from the circular hole. The effect of the hole on the displacement and
stress fields fades away from the boundery of the hole. This agrees well with the
predictions of the theory of elasticity (Hoek and Brown, 1980, pp. 103-109).
From the theory of elasticity, the radial and tangential stresses around a
circular hole in an infinite plate (Hoek and Brown, 1980, p. 104) are given by
Displacement distribution around a circular bole in a plate. The plate is subjected to uniaxial compression in tbe vertical direction.
t \ \ \
\ t \
\ +
+ +
1 • 71 ~ 1 02 P seAL
Figure 3.13 Principal stress distribution around a circular hole in a plate. The plate is subjected to unia:<.ial compression in the vertical direction . .... ," Tensile stress Compressive stress
--- -~. ,'" ~ .. ~.,-... . " - ,', .. - , ...
80
Where:
pz = applied vertical stress;
Ph = applied horizontal stress;
a = radius of the circular hole;
k = Ph pz
81
(3.43)
(3.44)
(3.45)
(), r = polar coordinates of the point. () is measured clockwise from the
vertical;
(7 r= radial stress;
(78 = tangential stress.
Figure 3.14 compares tangential stresses from the theory of elasticity with those
from FEM along three radial segments. These segments marked as AB, CD, and
EF on Figure 3.11. They make clockwise angles of 86.849,48.154, and 3.151 degrees
from the vertical respectively. The FEM tangential stresses correlates fairly well
with those of the elastic solution. However, more mesh refinement or use of higher
order elements will give more accurate FEM results. The deviations of the FEM
stresses from the theory of elasticity are larger along segment CD, the farthest from
the symmetry planes. The deviations also increase close to the hole boundary.
c C C ell
c C It) -
1.0
Figure 3.14
--- . -_ ..... _,.-.-.
1.5
o o
• •
2.0 Ria RATIO
•
82
AB
CD
EF
2.5
Comparison of tangential stresses around a circular hole in a plate. The plate is subjected to uniaxial compression in the vertical direction. Circles and triangles are FEM stresses. Solid lines are exact solution. AB radial segment at an angle or 86.849° irom ,'ertical. CD raclial segment at an angle of 48.154° from vertical. EF radial segment at an angle of 3.151° from vertical (Figure 3.11).
83
3.3.3 Single Edge Crack in a Plate Under Uniaxial Tensile Stress
A plate, of width b = 0.5 m, height h = 0.5 m, and thickness t = 0.1 m,
containing a single edge crack is modelled twice. In the £rst model, quarter point
(QQUAD8) and eight noded (QUAD8) elements are implemented. This model is
analyzed using 2 by 2 and 3 by 3 Gauss quadrature orders. In the second model,
only QUAD4 elements and Gauss quadrature of 2 by 2 order were employed. Due to
symmetry around the crack axis, only half the plate is modelled. Figure 3.15 shows
the geometry of the plate and the crack. The mesh for the modelled half is shown
in Figure 3.16 for the model employing QUAD4 elements. Figure 3.17 illustrates
the mesh of the model using QUAD8 and QQUAD8 elements.
Figure 3.15
Q-l
b
Single ed~e crack in a plate subjected to uniaxial tension. lafter Banks-Sills and Sherman, 1986)
One hundred elements are used for both models. The dimensions of the
mesh, the dimensions of the elements, as well as those of the crack are kept the
same in both models. A tensile stress of 2 x lOS N 1m2 is applied to the plate in
the vertical direction as shown in Figure 3.15. The crack has a length, a, half the
width, b, of the plate. The tip of the crack is located at
Figure 3.16 Mesh models single edge crack in a plate subjected to uniaxial tension. The plate modelled using QU AD4 elements. Dimensions are in meters, thickness is 0.1 m. Crack tip is at point T.
Mesh models single edge crack in a plate subjected to uniaxial tension. QUADS and QQUAD8 are implemented. Dimensions are in meters, thickness is 0.1 m. Crack tip is at point T.
---_ ... -- . .. ._ ... _ .... ~ ..... - -'"
86
point T (Figure 3.16 and Figure 3.17) and the crack face extends to the left. The
plate material has the elastic properties of Lithonia granite (Lama and Vutukuri,
1978, p. 385), E = 10.41 X 109 N 1m2 and p. = 0.19. The geometry of the problem is
chosen similar to the geometry of a single edge crack problem solved by Banks-Sills
and Sherman (1986) for the purpose of comparison.
The stress intensity factor, K1, due to the presence of the crack, can be
calculated using the FEM displacements of the nodes along the crack surface. The
displacement components needed are those normal to the crack surface. Two meth
ods are available (Banks-Sills and Sherman, 1986). The first method only makes
use of the displacements of the nodes of the QQUAD8 (quarter point) element along
the crack face. The formula used is:
(3.46)
The second method (called displacement extrapolation method) uses the displace
ment components normal to the crack face for all nodes along the crack face except
the nearest node. The method uses the equation:
(3.47)
Where:
K I ,Kj = stress intensity factorj
VB = displacement of node B (dislocated node of QQUAD8)j
Vc = displacement of node C (the third node of QQUAD8 from the tip)j
nodes B and C are shown in Figure 3.3bj
R. = length of the quarter point elementj
l'<r) = displacement component normal to the crack face at distance r from
the tip along the crack facej
r = distance from the crack tip along the crack surfacej
K = constant= ~~~~~ for plane stress, and = (3 - 4Jl) for plane strain.
____ " _~ , " 'M"-_'~" _ ',.-' ___ , _ •
87
Values of ria are plotted against Kj. The intercept of the correlation line with
the Kj axis at r=O would give the stress intensity factor. The values of the calcu
lated stress intensity factor are normalized by dividing it by qv:;a to find the non
dimensional stress intensity factor, which can be compared with results for similar
geometries. q is the applied tensile stress.
Displacements and principal stresses are shown in Figure 3.18 and Figure
3.19 for the model using QUAD4 elements. The distributions for the model using
QQUAD8 and QUAD8 elements are illustrated in Figure 3.20 and Figure 3.21.
The scales on all the figures of displacement and stress distributions represent the
maximum value of the plotted variable. All values are normalized with respect to
the maximum before being plotted. From the stress distributions, we can see the
high stress concentrations near the crack tip, about 15 times the applied stress (the
applied tensile stress is 2 x 105 Nlm2 ). The effect of the presence of the crack on
the magnitudes and directions of the stresses extends to distances roughly equal to
the length of the crack. The effect is more significant above the tip of the crack. Kj
and ria have been calculated and are plotted in Figure 3.22. The stress intensity
factors are calculated from these plots and using equation 3.46 as well.
The exact value of K/ for this geometry is 2.818 (Banks-Sills and Sherman,
1986). The calculated values compare well with the corresponding results obtained
by Banks-Sills and Sherman(1986). They reported a percentage error of -0.78 for
the displacement extrapolation method and of 8.8 for values obtained from equation
(3.46). The corresponding percentage errors obtained from our models are -.07
and 8.41 respectively. The percentage errors obtained from the present study are
summarized in Table 3.4.
----_. -~. . .. .-.. ~ .... ~ . ,""- -...
88
METERS
Figure 3.18 Displacement distribution through a plate containing a sin~le edge crack. The plate is modelled using QUAD4 elements (FIgure 3.16).
* I' t t t, t t t t t t " t t t t t
* tt tt""" If " II:t t
t t\ \\ ,\ \, It tl IIII '"
'" \\ \\ \, t, fllllll' f
'" \\ \\ \\ if flllll'" ~ \' \\ \\ \\ \t il 111'" #
.. ~\ \\\\\\\ttlllll-.~
..... ~,\\\\\ /1/lllu~
.... ~ ,,~ -, ~ ~ /.11// ~ ~ ~ . ~ - I \ Jt', •
89
Figure 3.19 Principal stress distribution through a plate containing a sin~le edge crack. The plate is modelled using QUAD4 elements {Figure 3.16).
Figure 3.20 Displacement distribution through a plate containing a single edge crack. The plate is modelled using QQUADS and QUADS elements (Figure 3.17).
• • • • • • • • • • t t t t f f f • f • ... • • • • • • · , • • • • , , f , • • ... • •• , , , , , ,
• t • f , , , , ,. . IS. , , , , , , , t A , , , , , , . I , \ , , , , , \ , \ ,t , , I , , I · , .-\ ~ , · , , \ \ \ \ i I I I I I , · , MI
... , ... " . ~\ \\\\ 1111 II .~ ~
.. .-+ ~- ,,., I f ;t~ " .~ + ~ I 1 . I t . .. ~ ., -+
3.24MIO & PASCAL
Figure 3.21 Principal stress clistribution through a plate containing a single edge crack. The plate is modelled using QQUADS and QUADS elements (Figure 3.17).
F' 3?? Plots using displacement extrapolation method to calculate the 19ure .--stress intensity factor. (a) Model using QUAD4 elements. (b) Model using QUADS and QQUADS elements and 2 x2 quadra-ture order. (~model using QUADS and QQUADS elements and 3 x3 qua rature order .
The crushed zone is a result of interaction between rock and explosive load
ing. It is preferred to use dimensionless relations to relate rock and explosive prop
erties to each other. These dimensionless relations have the advantage of making
it possible to compare results from tests with different explosives in different rocks.
The following ratios are defined:
diameter of crushed zone (Der) Scaled crushed zone diameter = -~---~-=--~~~~--:---'
diameter of blasthole (Db.h.)
characteristic impedance of explosive Characteristic impedance ratio (Z) = --------=~-----=---
characteristic impedance of rock
" I' . _ velocity of detonation (VOD) ve OClty ratIo - ck I . d' al I . (C) ro ongltu 10 wave ve OClty p
M d· . medium stress (O'm)
e mm stress ratIO = ------.--....:......;.;.;.:;~~ rock compressIve strength (O'e)
detonation pressure (Pd) Detonation pressure ratio = -~-----:--=------=-:~:.........:
rock compressive strength (0' c)
The data used here is obtained from reports published by the United States
Bureau of Mines. These reports include studies to compare relative performance of
explosives in granite (Atchison and Tournay, 1959; Atchison and Pugliese, 1964 b;
-----. -~ ,.,. '-"~""'-'
98
Nicholls and Hooker, 1965), in salt (Nicholls and Hooker, 1962), and in limestone
(Atchison and Pugliese, 1964 a). Tests were perfonned in joint free homogeneous
rocks. These studies used vertical blastholes. The explosive charges were placed,
stemmed, M.d detonated at the bottom of the holes far from any free surfaces. Hole
depths ranged from 10 to 26 feet. After a charge was detonated, the crushed rock
was blown from the blasthole by compressed air. The crushed zone volumes were
measured by adding sand, in small increments of known volume, into the cavity.
The height of the sand was measured between sand additions until the cavity was
filled.
In some of the referenced reports, repeated shots were carried out in the
same hole. Only crushed zone volumes from the first shots are taken into account
in the present analysis. The analysis investigates the effect of the velocity ratio,
characteristic impedance ratio, medium stress ratio, and detonation pressure ratio
on the scaled crushed zone diameter.
4.1.1 A Model for the Crushed Zone Geometry Around a Cylindrical Charge
The geometry of an explosive charge in rock is usually cylindrical. The
crushed zone boundary formed by such a charge can be modeled as a cylindrical
surface around the original charge and two hemispherical surfaces at the top and
bottom of the charge. The cylindrical and spherical parts have the same diameter.
Figure 4.1 illustrates the model.
The volume of the crushed zone, Vcr, is calculated as a function of its radius,
acr, and the charge height, H, using the following equation:
(4.4)
Equation (4.4) can written in the form of a third order polynomial as:
(4/3 7r )a~r + (7r H)a~r - (Vcr) = 0 (4.5)
If the crushed zone volume and the charge height are measW'ed, acr can be
found from equation (4.5) as a real positive root greater than or equal to the original
99
---- -------.-~~~~~
---- ---------~~~~~
Db.h . .. • Dcr , .. •
Figure 4.1 A model for the crushed zone around a cylindrical charge.
---- . -- . .. .- .. -.....
100
blasthole radius. Equation (4.5) is solved numerically, using the secant method
(Press et al, 1986, pp. 248-251) to find the crushed zone radius for each blasthole.
4.1.2 Effect of Explosive Properties on the Crushed Zone in Granite
Three sets of data are extracted from the studies performed on Lithonia
granite. Data set 1 is extracted from the tests by Atchison and Toumay (1959).
They used six types of explosives and two blasthole diameters, 3 1/16 and 4 1/16
inch. Charge height-to-diameter ratio ranged from 2.8 to 9.5.
Data set 2 is obtained from Atchison and Pugliese (1964 b). Five explosive
types were employed in blastholes with diameters of 1 1/2, 2 1/2, 3, and 3 1/2
inches. Charge height-to-diameter ratios ranged from 4.5 to 8.
The third data set is obtained from Nicholls and Hooker (1965). They used
blastholes of 3 inch diameter, charge height-to-diameter ratios from 1 to 1.9, and
six types of explosives. Data and calculations of the scaled crushed zone diameter
for the three data sets are given in Appendix B, Table B.1 through Table B.3.
A curve fitting program (Cox, 1985) has been used to calculate the correlation
factors. The program has the capability to calculate the correlation factors and the
corrected correlation factors for twenty five standard equations. According to the
values of the correlation factors, the program recommends the equation which best
fits the data set and calculates the constants of the recommended equation. Tables
B. 7 through B. 10, Appendix B, show samples of the curve fitted equations which
have correlation factors close to the correlation factors of the best fitted equations.
It can be seen that a number of equations have correlation factors close to that
of the best fitted equation for a given data set. The best fit equation is chosen
according to the highest correlation factor.
For the first curve fitting attempt, all the data from the three sets has been
pooled. The square of the correlation factors (R2) and the corrected correlation
factors (R~) are less than 0.3. The correlations are weak and the data points are
scattered over a wide range. Each data set is then checked and analyzed individually.
Data set 2 shows no correlation. Data set 1 shows R2-values greater than 0.5. Data
101
set 3 shows R2 -values greater than 0.8. Data set 2 is removed from the analysis.
Data for liquid oxygen (Table B.1) is removed from data set 1.
Data set 2 includes four different blasthole diameters and four different prim
ing charges. In addition, in some tests, the blasthole section directly above the
charge is of larger diameter than the charged section. All these factors contribute
to the weak correlations. Table B.2 shows large standard deviations in this data
set.
The liquid oxygen tests has been taken out of data set 1 because oxygen evap
orates from the carbon cartridges leaving some annular decoupling space between
the saturated cartridge core and the blasthole bOWldary. This causes less crushing
than completely coupled saturated cartridges and contributes to the scatter of data.
Data set 3 and data set 1 (excluding liquid oxygen tests) are analyzed
together. The best fitting relation between the scaled crushed zone diameter
(Dcr/Db.h.) and the velocity ratio (VOD/Cp) is:
(Dcr/ Db.h.) = [ ] A«VOD /Cp) + B)2 + C
1 (4.6)
Equation (4.6) and the data points are plotted on Figure 4.2, (a). The scaled
crushed zone diameter Dcr/ Db.h. increases from 1 at a velocity ratio of 0.5 to about
3 at a velocity ratio of 1.15. The rate of increase of (Dcr/ Db.h.) decreases beyond
(VOD/Cp) equal 1 and is almost 0 when (VOD/Cp) reaches 1.15. An increase
of (VOD/Cp) over 1.15 causes a slight decrease in (Dcr/Db.h.) . This decrease of
(Dcr/ Db.h.) is due the statistical treatment of data but physically is not feasable.
Lack of data beyond velocity ratio of 1.3 and the different number of tests at each
velocity ratio may contribute to the misbehaviour of equation (4.6) in the region of
velocity ratios greater than 1.2.
The. best fit relation obtained between (Dcr/ Db.h.) and the characteristic
impedance ratio, Z, is:
Z (Dcr/Db.h.) = (AZ + B) (4.7)
0.5
0.1
Figure 4.2
0.8
08
0.1 0.8 0.8 1.0 VELOCITY RATIO
(a)
00
08
I.'
0.1 O.S 0.4 0.0 0.8 CHARACTERISTIC IMPEDANCE RATIO
(b)
1.2
o o
Effect of the velocity ratio and the characteristic impedance ratio on the scaled crushed zone cliameter in granite. (a) Velocity ratio. (b) Characteristic impedance ratio.
10.0 110.0 40.0 10.0 10.0 '70.0 DETONATION PRESSURE RATIO
(b)
Effect of the medium stress ratio and the detonation pressure ratio on the scaled crushed zone diameter in granite. (a) Medium stress ratio. (b) Detonation pressure ratio.
- , ...... ,.. --I ., - •• , - .'_ •.
105
4.1.3 Effect of Explosive Properties on the Crushed Zone in Salt
Crushed zone data for blasting in Winnfield salt are extracted from Nicholls
and Hooker (1962). They tested four explosive types in blastholes with a diameter
of 3 inches. The ratio of charge height to charge diameter ranged from 1.9 to 3.7.
The relationship between the scaled crushed zone diameter and the velocity
ratio is found to have the form of equation (4.6). Constants for relationships of the
scaled crushed zone diameter in salt are given in Table 4.2. Equation (4.6) and the
data points for salt are plotted in Figure 4.4, (a). The scaled crushed zone radius
increases as (VOD/Cp ) increases. The rate of increase becomes much less at about
(VOD/Cp ) of 1.
The relationship between the scaled crushed zone diameter in salt and the
characteristic impedance ratio has the form:
(Dcr/ Db.h.) = [ 2] A(Z+B) +C
1 (4.10)
Equation (4.10) and data points for the characteristic impedance ratio are
plotted in Figure 4.4, (b). Increasing the characteristic impedance ratio increases
(Dcr/Db.h.) up to Z = 0.75. The rate of increase becomes smaller after the char
acteristic impedance ratio reaches 0.5. A slight decrease of (Dcr/Db.h.) is observed
beyond (VOD/Cp ) ratio of .75.
Table 4.2 Constants and Correlation Factors for the Equations Fitted to Predict the Scaled Crushed Zone Diameter in Salt.
O.S 0.1 0.7 0.8 ..1 CHARACTERISTIC IMPEDANCE RATIO
(b)
Effect of the velocity ratio and the characteristic impedance ratio on the scaled crushed zone diameter in salt. (a) Velocity ratio. (b) Characteristic impedance ratio.
16.0 111.0 ::16.0 46.0 60.0 D,ETONATION PRESSURE RATIO
(b)
Effect of the medium stress ratio and the detonation pressure ratio on the scaled crushed zone diameter in salt.(a) Medium stress ratio. (b) Detonation pressure ratio.
109
It has been decided to treat the data for the blastholes of 3 inch diameter
separately from the data for the 5 3/4 inch diameter holes. The relationship between
(Dcr/Db.h.) and (VOD/Cp ) is found to have the same form, but with different
constants, for the two data sets. The equation obtained was:
(4.13)
The relations obtained from the three correlations and the data points for the
dependency of (Dcr/Db.h.) on (VOD/Cp ) are plotted on Figure 4.6, (a).
The relationship between (Dcr/ Db.h,) and the characteristic impedance ratio
for the two data sets was found to have the form of equation (4.12). The equations
obtained from the three correlations along with the data points for limestone are
plotted in Figure 4.6, (b).
Table 4.3 Constants and Correlation Factors for the Equations Fitted to Predict the Scaled Crushed Zone Diameter in Limestone.
Equation No. Db•h• for Data A B C R2 R2 c
For Velocity Ratio:
(4.11) 3 and 5 3/4 inch 0.7442 -0.0786 0.03 0.03
(4.13) 3 inch 1.795 0.5760 0.22 0.02
(4.13) 53/4 inch 1.1540 0.8282 0.21 0.14
For impedance ratio:
(4.12) 3 and 5 3/4 inch 0.7192 -0.1049 0.07 0.01
( 4.12) 3 inch 0.5330 -0.06029 0.32 0.14
(4.12) 53/4 inch 0.8123 -0.1273 0.32 0.26
From Figure 4.6, we can see a slight increase in the scaled crushed zone
diameter with increasing velocity ratio. The scaled crushed zone diameters for 3
Effect of the velocity ratio and the characteristic impedance ratio on the scaled crushed zone diameter in limestone. Solid lines are for both 3 and 5 3/4 inch diameter blasholes; dashed lines and triangles are for 5 3/4 inch diameter blastholes; dotted lines and circles are for 3 inch diameter holes. (a) Velocity ratio. (b) Characteristic impedance ratio.
110
111
inch diameter holes (dotted lines) are about 30 % higher than for 5 3/4 inch diameter
blast holes (dashed lines). The number of data points for 5 3/4 blastholes is twice
the number of data points for 3 inch holes. This causes the regression curves of the
pooled data (solid lines) to be closer to the dashed lines. Atchison and Pugliese
(1964 a, p. 17) point out that cleaning of larger blastholes by compressed air may
be incomplete. The capability of the compressed air to lift crushed rocks is less than
in small holes. This causes underestimation of the volume of the crushed zone.
Disturbance caused by the compressed air can loosen coarse fragments from
the fractured zone, outside the crushed zone, into the cavity. This partial filling of
the cavity may contribute to the cavity volume underestimation. First it leads to
underestimation of the height of the cavity. In addition, bridging of these coarse
fragments can prevent compressed air from completely removing the crushed rock
and prevent sand from completely filling the cavity. Consequently, the measured
cavity is less than the actual one.
4.1.5 Effect of Explosive Properties and Rock Properties on the Crushed Zone
It has been shown that the scaled crushed zone diameter increases at a de
creasing rate with an increase in velocity, characteristic impedance, medium stress,
and detonation pressure ratios. Rock strength increases with increase in load
ing rate. Using explosives of high VOD means loading rock at a high rate and
causes higher rock resistance to the explosion pressure. This explains why the
(Dcr/Db.h.) ratio increases at a decreasing rate when (VOD/Cp ) is increased. In
each rock, tests have been published over a limited range of the ratios.
The relations obtained between the scaled crushed zone diameter and velocity
ratio in granite, salt, and limestone (for 3 inch diameter holes only) are plotted
together in Figure 4.7, (a). The relations obtained between the scaled crushed zone
diameter and the characteristic impedance ratios for the three rocks are plotted
in Figure 4.7, (b). From Figure 4.7, (a), for a velocity ratio from 0.5 to 2.2, the
scaled crushed zone diameter increases with increasing velocity ratio up to about 1.
Detonation velocity ratios greater than 1 only slightly increase the scaled crushed
D.B . D.' D.' I.' I.S CHARACTERISTIC IMPEDANCE RATIO
(b)
I.S
Effect of the velocity ratio and the characteristic impedance ratio on the scaled crushed zone diameter in rocks. (a) Velocity ratio. (b) Characteristic impedance ratio.
112
113
zone diameter. The rate of increase of the scaled crushed zone diameter is higher
in rocks with higher compressional wave velocity. In general, rocks of higher Cp
have larger scaled crushed zone diameters. The correlation factors for limestone are
not good. However, the relations obtained may support the trend of insignificant
change in (Dcr/Db.h.) as the (VOD/Cp ) ratios extend beyond the range studied
in granite and salt.
Figure 4.7 (b) shows that rocks of higher characteristic impedance have a
larger scaled crushed zone diameter at a given characteristic impedance ratio. The
rate of increase of the scaled crushed zone diameter is higher for rocks of higher
characteristic impedance. The rate of increase in all the three rocks decreases with
increasing characteristic impedance ratios.
Relations between the scaled crushed zone diameter and medium stress ratios
for granite and salt are plotted in Figure 4.8, (a). Relations between scaled crushed
zone diameter and detonation pressure for these two rocks are plotted on Figure
4.8, (b). Both figures show a decreasing rate of increase of the scaled crushed zone
diameter as the medium stress and detonation pressure ratios increase. The rate of
increase is higher in granite than in salt.
Figure 4.8 (b) has the potential to be used to estimate the dynamic strength
of a rock. A scaled crushed zone diameter of 1.0 means no crushing. The corre
sponding point obtained by the relations on the detonation pressure axis can be
considered as an estimate for the dynamic strength of the rock. From the figure
the dynamic compressive strength of granite is about 9 times its static strength.
For salt, as mentioned before, the employed strength is a dynamic strength. The
estimated ratio of two is not the real dynamic to static strength ratio. Employing
the static strength has the effect of moving the curve to the right. The real ratio
would be larger.
The scaled crushed zone diameter varies from one rock-explosive combina
tion to the other. Within the range of ratios studied, (Dcr/ Db.h.) attained a max
imum of about 1.8 in salt, 2.2 in limestone, and 4.0 in granite. The variation of the
scaled crushed zone diameter should be taken into account. In numerical modeling,
it can affect the results. In smooth blasting, explosives which give (Dcr/Db.h.) close
----'. --. , ... -.---'.,
o c:" ~
; ~ granite < - - - - salt -Co ~. z .. C No CN ~ = CIlo ~.
c:" CJ c., _----. "-2': - .. -c50 ~~+------r----~------T-----~----~------~----------~
Effect of the medium stress ratio and the detonation pressure ratio on the scaled crushed zone diameter in rocks. (a) Medium stress ratio. (b) Detonation pressure ratio.
---_ .. -- .... _ .. -- •.... "'
115
to 1 will cause less disturbance around blastholes.
In underground blasting, spacing of blastholes in burn cuts should not be
less than the crushed zone diameter. This will prevent charge blowouts.
4.2 Shape of Blasting Wave Fronts in Bench Blasting
In this section, the effect of the (VOD/Op ) ratio on the shape of the wave
front is investigated.
4.2.1 Construction of the Wave Fronts at Different Velocity Ratios
The velocity of detonation (VOD) of commercial explosives has a wide range.
The VOD ranges from 2,103 m/sec (6,900 ft/sec) for permissible ammonia dyna
mite to 7,010 m/sec (23,000 ft/sec) for nitrogen tetroxide and kerosine. The VOD
increases with an increase in charge diameter up to some critical diameter. Stronger
explosives attain their maximum VOD at smaller diameters than weaker explosives.
For example, straight gelatine 60% HE attains its maximum VOD at 5.1 em (2 inch)
diameter while ANFO achieves its maximum VOD at 17.8 cm (7 inch) diameter
(Dick et al, 1983, p. 14). Also, rocks have a wide range of compressional wave
velocity. Dunite is an ultra-basic plutonic igneous massive strong rock. Dunite has
a compressional wave velocity of 7 km/sec (22,989 ft/sec) while coal has a compres
sional wave velocity as low as 1.2 km/sec (3,941 ft/sec) (Lama and Vutukuri, 1978,
p. 240). Hence, the velocity ratio for an explosive-rock combination can range
widely. For nitrogen tetroxide and coal, the velocity ratio is 5.8. For dunite and
permissible ammonia dynamite combination, the velocity ratio is 0.3.
The shape of the wave front generated by a vertical cylindrical charge is
constructed for a variety of (VOD /Op) ratios. These ratios include infinity, greater
than one, equal to one, and less than one. Huygen's principle (Telford et al, 1976,
pp.243-244) is used for the construction of the shape of the wave fronts. According
to Huygen's principle, every point on the wave front can be regarded as a new
source of waves. The wave fronts are constructed only for the initial outgoing
compressional waves. It is assumed that these waves are propagating in an isotropic,
116
homogeneous, and linearly elastic rock medium. Reflected waves are not shown for
the sake of simplicity and clarity. The explosive charge is assumed to be continuous,
fully coupled, stemmed, bottom initiated, and detonating at a constant speed in all
cases.
The geometries of the charge and the bench for all constructions are kept
the same. Dimensions of the bench and the explosive charge are as follow:
Blasthole diameter, Db.h = 10.0 cm,
Rock burden, B = 3.0 m,
Bench height, H = 9.0 m,
Stemming length, ST = 2.0 m,
Overdrilling length, OV = 1.0 m,
Charge height, Hch = 8.0 m.
In all figures, solid wave fronts represent the last position and shape of the
outgoing wave front after an arbitrary propagation time. Dashed wave fronts repre
sent the shape and position of the outgoing wave front at intermediate times between
the instant of initiation and the arbitrary time of the last position. Elapsed time,
T, is given in milliseconds after initiation.
Figure 4.9 shows the shape of the outgoing wave front for an infinite velocity
ratio in a rock of 3,000 m/sec compressional wave velocity. Practically this assump
tion is not valid but it is useful as an upper limit for the velocity ratio. At the
top and bottom of the blasthole, the wave front is hemispherical. Between the top
and the bottom the wave front has a cylindrical surface. The minimum angle of
incidence is zero. The minimum angle of incidence refers to the angle of incidence
in the vertical plane which is perpendicular to the bench free face and contains the
centerline of the blasthole. The propagating wave front intersects the free face at
points diverging from the symmetry plane and symmetrically distributed around
it. When the wave front propagates as a right cylinder, these symmetrical points
of incidence lie on the generators at which the wave front intersects the free face.
As the vertical plane containing the angle of incidence diverges from the symmetry
plane, the angle of incidence increases. This is important to recognize because the
Figure 4.9
---- . -~ . .. .- .. -.. ~. -.
I I
I I
I ,
BENCH TOP .... .. , .... , , .. , ,
I I I I
I I , I ,
~
~
,I ,
.-I.
-.. ", .... , ...., .... " , , , \
"'I. \, , " ,
, , , , ,
I I I I , , , , :
" ''I. " " , \ , .... __ '" I I , \ \ I I ' , \ , , I I , ... ' , I , , ' '.... ..." ,
, " ... -.-- " I \ ....... __ .... ' ,
, I , ' , ' " " .. " ............ ...-' -------
117
~ 0 ~ ~ ~ = ~
::: 0 Z ~ =:I
FLOOR
Shape of the outgoing wave front for an infinite velocity ratio. Wave front positions 1, 2, 3, 4, and 5 are at elapsed times of 0.25, 0.5, 0.67, 1.0, and 1.5 msec respectively.
118
wave fronts are not planar within the range of the rock burdens used in bench
blasting.
Figure 4.10 shows the shape of the outgoing wave front for a velocity ratio
of 3. In this example, VOD is 4,000 m/sec and Cp is 1,333 m/sec. The shape of the
wave front is spherical below the initiation point (bottom of the blasthole). Above
the bottom of the blasthole, the wave front propagates as a conical surface until
the detonation front reaches the top of the explosive column. Then, the wave front
propagates spherically in the top portion of the bench. This geometry of the wave
front can be referred to as a sphero-conical shape. The minimum angle of incidence
of the wave front is 19 degrees.
The shape of the outgoing wave front for a velocity ratio of 2 is shown in
Figure 4.11. The rock has a Cp of 2,000 m/sec and the explosive has a VOD of
4,000 m/sec. The shape of the wave front is sphero-conical. The minimum angle
of incidence is 38 degrees.
Figure 4.12 shows the shape of the outgoing wave for a velocity ratio of 1.5.
Here, the von is 4,000 m/sec and Cp is 2,667 m/sec. The shape of the wave front is
sphero-conical. The minimum angle of incidence is 41 degrees. From Figures 4.10
through 4.12, we can see that with a decrease in the velocity ratio the minimum
angle of incidence at the free face increases. The conical part of the wave front
decreases while the spherical part increases.
In the conical part of the wave fronts, the minimum angle of incidence is
equal to one half of the apex angle of the cone. The sine of this angle is equal to
the ratio of the distance travelled through the rock to the distance travelled along
the charge column at a given time, or simply the reciprocal of the velocity ratio.
This is true for all velocity ratios greater than one and less than infinity.
The shape of the outgoing wave front for a velocity ratio of one is shown in
Figure 4.13. The rock Cp and the explosive VOD are equal and have a magnitude
of 3,000 m/sec. The shape of the wave front is completely spherical and the center
for this propagating sphere coincides with the initiation point. The minimum angle
of incidence at the free face, is about 19 degrees at the toe and is about 77 degrees
Shape of the outgoing wave front for a velocity ratio of 3. Wave front positions 1,2, 3, and 4 are at elapsed times of 1.13, 1.97, 2.38, and 3.44 msec respectively.
120
BENCH TOP
...... " , ,
I . , , , I ,
I ,
I , ~ I I , I 0 I I ..
I I " ~ , .. I
, " I
" , " I I .. ~ I I , I I I
, " ~ I I I \ , " I I I I , " ,
I I I I , " , ~ , I I I " , , , I I " " I I , , " , " == , , , I ""
Figure 4.11 Shape of the outgoing wave front for a velocity ratio of 2. Wave front positions 1, 2, 3, 4, 5, 6, and 7 are at elapsed times of 0.65, 1.50, 1.60, 1.97, 2.77, 2.96, and 3.47 msec respectively.
BENCH TOP
, , , , , , , ,
, , , ,
, ,
, , , ,
~. , , , , , , , ,
..
, ...
" " , , , , ..
... ... ... ..
.. .. ...
... ... ... ...
.. ... ... ..
... ... ... ... ... ... ..
" , , .... I' .n.,~ ", .... .. ...
" " ~ " " .. , " .. ... .. ... ... ..
...
" " " .... ... , , ..
, I' .. , " .... , " .. , " \
... ..
.. .. .. FLOOR
, " \ I " ,'. , " .'. I " .'. I • I .'. I I I ,I I I • I ,'. · " ,'. · ., .', · ,\ ,', · ,\ ,', , \ \ I',
\ .. I ' , .... I I , \ .... , I , \ ...... ' I , \ .... " , '.. .... ........ ........,' "
.. ... ... - ••• --.., I .. .... ..,I I , «lit... .. I , .............. -_....... , .. ,
Figure 4.12
.. , .. , ... , ...... ' ........ , .. "
flit... fill' '- ",--- .. --.... _----_ .. -_ .. -Shape of the outgoing wave front for a velocity ratio of 1.5. Wave front positions 1, 2, 3, and 4 are at elapsed times of 1.01, 1.17, 1.99, and 2.68 msec respectively.
Figure 4.13 Shape of the outgoing wave front for n velocity ratio of one. Wave front positions 1, 2, 3, 4, 5, and 6 are at elapsed times of 0.50, 1.00, 1.50, 2.00, 2.50, and 3.00 msec respectively.
123
Figure 4.14 shows the outgoing wave front for a velocity ratio of 0.8. The velocity
of detonation is 4,000 m/sec and Cp is 5,000 m/sec. When the velocity ratio is
less than unity, the wave front moves faster in the l'ock than the movement of the
detonation front along the explosive colunm. The separation between the initial
wave front and the detonation front increases with an increase in. the elapsed time
from the instant of initiation. The spherical wave front shown in Figure 4.14 is for
the pulse generated at the initiation point. Inside this wave front are a series of
continuous wave pulses lagging behind it. It is not possible to construct an envelope
to represent a unique wave front for all pulses, because each point in the explosive
column creates a wave in which the detonation front lags behind the wave motion
in the rock. This situation elongates the duration of the pulse experienced by each
point in the rock.
A rock particle, at a given point around the charge, experiences motion for
a certain period of time after the arrival of the wave front and before the arrival
of the largest amplitude contribution to the particle motion. This period of time
(amplitude lag) depends on the shape of the propagating wave front and on the
relative position of the point with respect to the cylindrical charge.
Around the axis of the cylindrical charge, the amplitude lag increases with
increasing distance from the charge. If the wave front is cylindrical, the largest
motion amplitude accompanies the wave front and the amplitude lag is minimum
and is equal to the rise time. ruse time is defined as the time between the first motion
arrival and the peak arrival. For a conical wave front, the amplitude lag increases
with decreasing (VOD/Cp ) ratio. A decrease in the (VOD/Cp ) ratio continues to
cause increase in the amplitude lag for the spherical wave fronts produced when the
(VOD/Cp ) ratio becomes equal to or less than one. The amplitude lag is equal to
the rise time for the vertical amplitude component and is less than the rise time for
the horizontal radial component. When the (VOD/Cp ) ratio is less than one, the
amplitude lag becomes larger compared to those for (VOD/Cp ) ratios equal to or
greater than one. For velocity ratios equal to or less than one, the amplitude lag
depends not only on the distance from the charge but also on the distance from the
point of initiation. The amplitude lag increases with increasing distance
Shape of the outgoing wave front for a velocity ratio of 0.8, Arrows along the charge colullUl show the positions of the detonation front. Wave front and detonation front positions 1, 2, and 3 are at elapsed times of 0.61, 1.20, and 1.80 msec respectively.
125
from the point of initiation. In Figure 4.14, for example, point A at the bench
face is 1.5 m above the toe and 3.75 m from the point of initiation. The largest
amplitude contribution (represented by the dotted circle) arrives at point A after
an elapsed time of 1.22 milliseconds. This elapsed time is the sum of travel time
along the charge column at a speed equal to VOD, from the point of initiation to
point 1, and the travel time from point 1 to point A, at a speed of Cpo Point A
has been in motion since an elapsed time of 0.75 millisecond, travel time from the
point of initiation to point A at a speed of Cpo This gives an amplitude lag of 0.47
millisecond.
Below the initiation point (bottom of the charge), the shape of the wave front
is spherical for all the (VOD ICp ) ratios. In this region, for all the (VOD ICp ) ra
tios, the largest amplitude contribution accompanies the wave front and the am
plitude lag is minimal. For an infinite velocity ratio, the amplitude lag is equal to
the rise time and the duration of the motion is short. For velocity ratios less than
infinity, the amplitude lag is less than the rise time and contribution to the motion
amplitude is attained from the entire charge length. The rise time and the peak
amplitude increase with increasing charge length.
In the stemming region, the characteristics of the particle motion also de
pends on the velocity ratio. For an infinite velocity ratio, the wave front is spherical
and the amplitude lag is equal to the rise time. For velocity ratios greater than one,
in the spherical part of the wave front, the largest amplitude contribution accom
panies the wave front and the amplitude lag is equal to the rise time. For velocity
ratios equal to one, the amplitude lag is equal to the rise time. For velocity ratios
less than one, the amplitude lag is less than the rise time and the largest amplitude
contribution does not accompany the initial wave front.
These wave front shapes have been constructed for a single column charge
initiated at the bottom. They are the fundamental wave front shapes. The same
procedure and principle can be used to construct the wave front shapes for decked
charges and charges with several initiation points with different locations or different
The wave front constructions show that there is a difference between the
motion characteristics resulting from explosive detonations with velocity ratios equal
to or greater than one and the motion characteristics resulting from velocity ratios
less than one. When the velocity ratio is equal to or greater than one, the amplitude
lag time of a point around the cylindrical charge is shorter than the amplitude
lag time for velocity ratios less than one. This means longer pulse duration for
(VOD/Cp ) less than one. An increase in pulse duration enhances the weakening
and/or fracturing capability of a stress pulse. The largest amplitude contribution
to the motion comes from the nearest explosive charge point. Contributions to
motion amplitude and amplitude lag from the explosive charge below the nearest
point of the charge depend on the (VOD/Cp ) ratio. For a (VOD/Cp ) ratio less
than or equal to one, a contribution is obtained from the entire charge between
the closest charge point and the initiation point. For a (VOD/Cp ) ratio greater
than one, contribution comes from a certain length of the explosive charge. This
charge contributing length increases with a decrease in the (VOD/Cp ) ratio and
with increasing distance from the charge. For all the (VOD/Cp ) ratios except
infinity, contribution to the motion amplitude and duration continues to come from
the rest of the explosive charge above the nearest charge point until the the pulse
from the top of the charge arrives at the point.
The incident compressive waves at the free face are reflected as shear and
tensile waves. If these tensile stresses are greater than the tensile strength of the
rock, tensile cracks form and the rock fails. The tensile cracks are perpendicular
to the direction of the tensile stresses creating them. This means that tensile crack
surfaces follow the shape of the reflected tensile wave fronts. For a velocity ratio of
infinity, the tensile crack surface follows the reflected cylindrical wave front surface.
For a velocity ratio greater than one, the tensile crack surface follows the reflected
conical wave front surface. For a velocity ratio equal to or less than one, the
tensile crack surface follows the reflected spherical wave front. In all these cases,
the surfaces of the tensile cracks will not isolate discrete blocks from the burden
----" -- , " ,",---,'
127
rock mass. There is a potential for the cracks coming from the top of the bench to
intersect the cracks propagating from the bench face and isolate the rock slabs, if
any, at the top of the bench. This can take place if a high Pel. explosive and small
burdens are implemented to produce large pulse amplitude at the free face. At the
bottom of the bench, the outgoing wave fronts have less potential to reflect and
form transverse cracks to separate the burden rock at the bench floor because of
the lack of appropriate free faces. Exceptions may occur if there are pre-existing
bedding or jointing planes.
Traces of the cracks at the top of the bench should be concentric circles for all
the (VOD IGp) ratios. This is because of the spherical wave front in the stemming
section and because the intersections between the reflected waves and the bench top
surface are circular for cylindrical, conical, and spherical waves. At the bench free
face, traces of crack surfaces are not the same. Cylindrical crack surfaces produce
vertical line traces; conical crack surfaces produce hyperbolic traces, and spherical
crack surfaces produce circles or circular arcs. These traces are symmetrical around
the projection of the blasthole at the bench free face. The hyperbolic and circular
arc traces are open downward.
At the free face, the highest stress level of the propagating wave develops
when the wave first touches the face along the projection of the blasthole on the
face. At this location, the travel distance and the angle of incidence of the wave . '
are minimal compared with the subsequent intersection locations between the prop
agating wave and the face. If the wave is strong enough to overcome the tensile
strength of the rock, the earliest cracks form before the formation of any spalling
cracks by the reflected waves. These cracks are vertical and their tips move inward
and upward in the rock mass. Small scale tests show the existence of these cracks
and mathematical treatment of the incident and reflected wave at the face show
that they take place for all wave front shapes (Wilson, 1987, Ch. 3 and Ch. 4). In
addition, Wilson (1987, Ch. 4) shows that the spherical wave fronts causes radial
and tangential cracking at the free face. Spalling crack tips advance into the burden
rock following the reflected wave fronts while the general direction of the impending
wave front is upward along the face. Hence, as the wave front moves upward, the
128
previously formed spalling cracks stop and new spalling cracks are initiated after
some travel distance along the face. This is because when a crack forms, it causes
stress relief and energy consumption in its neighborhood preventing new cracks from
being initiated.
At the top of the bench, if the amplitude of the reB.ected tensile stresses
is greater than the tensile strength of the rock, the reB.ected wave fronts have the
potential to form discrete rock blocks that separate from the rock mass. This
potential separation from the rock mass takes place because the formed cracks
daylight at the top of the bench. Some of the incomplete crack surfaces may be
completed later by the gas pressures. This· situation at the collar of the blasthole is
similar to what has been observed in crater tests (Duvall and Atchison, 1957). The
width of these discrete blocks increases as the velocity ratio decreases and the length
of the stemming increases. The depth of these fractures depends on the magnitude of
the tensile stresses in the reB.ected pulses and on how deep they maintain magnitudes
greater than the rock tensile strength. This depends on the original pulse generated
by the explosive, the attenuation coefficient of the rock, and the stemming length.
Shorter stemming results in faster crack initiation, higher stress levels, and deeper
cracking because of the shorter travel path of the waves. Hence, too short stemming
is undesirable because it allows early escape of the explosion gases. This results
in poor fragmentation and displacement of the fragmented material, B.y rock and
air blast problems. Too long stemming also is undesirable because it causes poor
fragmentation in the stemming region. From field practices, stemming lengths from
two thirds to one times the burden have proved to be successful in avoiding these
problems.
Figure 4.15 shows the amplitudes of the reB.ected dilatational and distortional
plane waves at different angles of incidence for v = 1/3 (Kolsky, 1973, pp. 28-
29). The amplitude of the reB.ected dilatational wave is equal to the amplitude of
the incident dilatational wave only at angles of incidence 0 and 90 degrees. The
amplitude of the reB.ected dilatational wave decreases with increasing the angle of
incidence to a minimum of 38% of the incident amplitude, at an angle of incidence of
65 degrees. When the angle of incidence increases beyond 65 degrees, the reflected
,. 1
H)
8
;
s
s
4
3
Z
o 1
0
Figure 4.15
129
I I ... -- ..... ... ", -............ "
..
'" ,
I
I I I
, ~ , , "'-/ I
, I , ,
/ ....... 1\\ / , ~ I
Y '\ 1 \ II \
\
,'/ I I~I )(
" I I I lL \
"" \
I \ , I
I \
I \ \ \ ,
I \ , \ \ I
-Az!A, -\ I AJ/A, --- \ , , I
I i , I .
o. . • • • • • • 10 30 ,40...J:' 'd~ Ang e Ql InC! enee .10 70 10
Amplitudes of reflected distortional and dilatational waves at different angles of incidence for II = 1/3. AJ, A2 , and A3 are amplitudes of incident, reflected dilatational, and reflected tortional waves respectively. (After Kolsky,1963, p. 29)
130
amplitude increases until it achieves 100% of the incident amplitude, at an angle of
incidence of 90 degrees. When the reflected dilatational amplitude is less than 100%,
the rest of the incident energy is reflected in the form of distortional waves. Keeping
in mind that wave energy is proportional to the square of the wave amplitude, the
reflected dilatational energy can be as low as 16% of the incident energy. The
minimum angle of incidence at the free face increases with decreasing velocity ratio.
Hence, the reflected dilatational amplitude decreases with decreasing velocity ratio.
This decreases the capability of the reflected waves to create cracks and/or extend
them.
The reflection theory (Duvall and Atchison, 1957 and Hino, 1959, pp.22-23)
postulates that the wave reflection mechanism is the main mechanism for burden
rock fragmentation and displacement. The theory ignores the gas pressure role and
claims that the rock is pulled apart, not pushed apart (Duvall and Atchison, 1957, p.
1). The above discussion of wave fronts partially agrees with the reflection theory in
the possibility of some crack and/or slab formation. However, at the bench free face,
the formation of discrete blocks is unlikely. Needless to say, more work is needed to
complete the formation of discrete blocks and to displace them. This means that
a real amount of work is left for the explosion gases. Explosion gases are expected
to complete rock burden separation from the rock mass by extending radial cracks
from the blast hole to intersect those at the free face, if any, or to intersect the free
face itself. Also gases are responsible for displacement of the fragmented rock.
The construction of wave fronts is useful in considering measurements of the
particle motions resulting from blasting. These measurements may be for study
ing relative performance of explosives or estimation of the attenuation coefficients.
The wave front constructions suggest that if the stress measuring instruments only
measure in the horizontal radial directions, they should be positioned at the level of
the initiation point. If the measurements are to be made at a different level, mea
surements of three orthogonal components is needed to completely and accurately
determine the displacement vector at that location. From the displacement vector,
other quantities can be calculated by differentiating with respect to time or dis
tance. The different wave front shapes and zones and their extension with distance
In this section a circular hole in a plate is analyzed. The objectives of the
analysis are: to get a goodassesment of the refinement needed close to the blasthole;
to find out what is the effect of representing the circular boundary by straight edge
or circular edge elements on the convergence of the solution; to see if using 3 x 3
Gauss quadrature order will give better convergence than 2 x 2 quadrature order;
and to find out to what degree the use of consistent loads can improve results
compared with lumped loads. Cook (1981, p. 406) states that using straight edge
elements to model circular boundaries causes strains and stresses normal to the
boundary to converge to the wrong values. He adds that errors in these strains and
stresses can be of the order of the tangential strains and stresses and he recommends
the use of elements with circular edges to avoid this error.
Hinton and Owen (1977, pp. 240 - 256) solve the problem of a thick cylinder
subjected to internal pressure under plane strain. They use 2 x 2 and 3 x 3 Gauss
quadrature orders. The same problem is solved here to compare the SABM program
results to their results. The problem is a circular hole of 5 m radius in a plate of 1 m
thickness. The outer circular boundary has a 20 m radius. The Young's modulus is
1000 N/m2 and the Poisson's ratio is 0.3. The internal pressure applied to the hole
is 10 N /m2. The eight noded isoparametric element is employed in the analysis.
The consistent nodal loads along a side of an eight noded isoparametric
element (quadratic edge) subjected to uniform traction (7y are calculated from the
surface integral:
(5.5)
The integration (Cook, 1981, pp. 10-11) gives the following consistent load vector
{r} = (711 t L{(1/6) (2/3) (1/6)} (5.6)
For linearly varying pressure (Bathe, 1982, p. 218), the consistent nodal loads are:
where:
[N] = matrix of approximation functions;
[4>] = vector of tractions along the element edge;
t = thickness of the element;
L = length of the side of the element;
PI, P2 = pressures at the end nodes of the element side.
139
(5.7)
Because of the symmetry, only one quarter of the problem needs to be ana
lyzed. Figure 5.1 ( a) shows the mesh in which the circular boundary of the hole
is modeled by circular edge elements. Figure 5.1 (b) shows the mesh in which the
circular boundary of the hole is modeled using straight line edge elements. Nine
elements are used in both meshes.
The elastic solution for the tangential and radial stresses for this problem is
given by (Jaeger and Cook, 1979, p. 137):
Where:
0" r and 0"8 = radial and tangential stresses;
RI and R2 = internal and outer radii;
PI = internal pressure;
r = radial distance from the hole center.
(5.8)
(5.9)
The radial and tangential stresses obtained from the Finite Element Method
(FEM), for 2 x 2 and 3 x 3 Gauss quadratures using the consistent loads, along
with the exact elastic solution are shown in Figure 5.2. The results obtained by the
SABM program are the same results obtained by Hinton and Owen (1977, pp. 241-
242). The differences between the FEM tangential stresses and the exact tangential
------.-~ . " '-"-.'"
Figure 5.1
0 40 C ..
:10
0 ori
S-ri :Ie
~ tl :17
:E~ o-f tl z ~ So
ori
o 0t:----__ -f~~~--~~~~,_--~e----~7
0.0 0.0 10.0 I~.O DISTANCE FnOM CENTEn. m
~O.O
(a)
o 40 :;i.....---
~ 7
1.0 10.0 10.0 10.0 DISTANCE FnON CENTER. m
(b)
Mesh for thick wall cylinder using nine eight noded elements. (a) Using circular edge elements along the hole boundary. (b) Using straight line edge elements along the hole boundary .
Radial and tangential stresses using 9 elements, lumped and consistent loads, 2x2 Gauss quadrature, and straight line segment modeling of the hole boundary.
e Q~--------~--'---~~~--~~--r--+------~
Figure 5.5
0.0 5.0 10.0 16.0 20.0 DISTANCE FROM CENTER, m
Mesh for thick wall cylinder using 18 eight noded elements. The hole boundary is modeled with circular edge elements.
Radial and tangential stresses using 18 eight Doded elements, lumped and consistent loads, 2x2 Gauss quadrature, and circular modeling of the hole boundary.
147
The problem is solved again using the 18 element mesh but the hole boundary
is modeled by straight line edge elements. The FEM radial and tangential stresses
along with the exact stresses are plotted in Figure 5.7. The FEM solution gives
the exact elastic radial and tangential stresses. Hence the statement of Cook (1981,
p. 406) that the stresses nonnal to the boundary converge to the wrong values if
the circular boundary modeled using straight edge elements is not valid when the
eight noded isoparametric element is used. Also his statement that the magnitude
of error is of the order of the tangential stresses is not valid even for the coarser
mesh of the 9 elements.
From the above analysis, some conclusions are drawn. Lumped loads, com
pared to consistent loads, give insignificant stress errors close to the boundary of
the hole for coarse mesh. The errors, for radial stresses, are about 5% at a distance
from the hole boundary of about 20% of the hole radius. Errors in the tangen
tial stresses are less than half the errors in the radial stresses. When the circular
boundary is modeled with straight line edge elements the errors are about double
the errors of circular edge elements. The difference in accuracy of stress calculations
between lumped and consistent loads diminishes at a distance approximately one
hole radius from the hole boundary. The 2 x 2 Gauss quadrature order gives
better convergence than the 3 x 3 quadrature order. Hence it is recommended
when the eight noded isoparametric element is employed. The 2 x 2 quadrature
order also demands less computation time. Modeling the circular boundaries with
straight line segments gives less accuracy in stress calculations than circular mod
eling of such boundaries. The stress errors caused by lumped loads and straight
line segment modeling of circular boundaries, vanish when the mesh is refined. The
solution converges to the analytical elastic solution. It is important to remember
that these conclusions are based on analysis with the eight noded isoparametric
Radial and tangential stresses using 18 eight noded elements, lumped and consistent loads, 2x2 Gauss quadrature, and straight line segment modeling of the hole bOWldary .
In this section and the following sections of this chapter, the material proper
ties of Lithonia granite are used in the finite element analyses. Lithonia granite has
the following properties (Atchison and Tournay, 1959, p. 3): compressive strength
of 2.0685 x (10)8 N/m2 (30,000 psi); tensile strength of 3.0275 x 106 N/m2 (450
psi); Young's modulus of 2.0685 x 1010 N /m2 (3 x 106psi). Nichols and Hooker
(1965, p. 3) report from their laboratory tests a Poisson's ratio of 0.19. The explo
sive material selected is semigelatine dynamite which has a detonation pressure of
6.3434 x 109 N/m2 (9.2 x 105psi), density of 1, 180.6kg/m3 (731b/jt3), and velocity
of detonation of 4,819 m/sec (15,800 ft/sec) (Atchison and Tournay, 1959, p. 24).
Hino (1959, pp. 61-63) explains the relation between the different explosive
pressures and the explosive specific volumes associated with them in the detonation
process. He postulates that for perfect explosive loading, the detonation pressure
at the end of the reaction zone (Chapman and Jouguet plane) is associated with a
specific volume less than the initial specific volume of the undecomposed explosive.
He adopts the detonation pressure for his shock wave analysis. When the gaseous
products of the detonation expand to the initial specific volume of the explosive,
the pressure drops to approximately half the detonation pressure. The pressure
at this stage is referred to as the explosion pressure. Accordingly, for the static
finite element analysis, the explosion pressure represents the upper bound of the
internal pressure applied to the blast hole wall. Because the semigelatine dynamite
is chosen, the upper bound of the internal pressure which can be applied to the
blasthole boundary is 3.1717 x 109 N /m2 •
The geometry assumed for the bench drilling pattern is a diameter of 10 cm,
burden of 3 m, spacing of 3 m, and bench height equal to or greater than three times
the burden. From the analysis of the crushed zone around a cylindrical charge in
chapter 3, the ratio of the crushed zone diameter to the blasthole diameter for this
rock-explosive combination is 2.84. Accordingly, in the finite element modeling,
the internal pressure is applied to a circular boundary of diameter 2.84 times the
150
nominal drilled hole diameter. For the 10 em diameter drill hole, the internal
pressure is applied to a crushed zone boundary of 28.4 em diameter.
Two dimensional finite element analysis of a plate of granite around the
blast hole is adopted. The plate is 10 em thick. It is assumed that Lithonia granite
is homogeneous, isotropic, and linearly elastic. Plane stress condition is assumed.
Figure 5.8 shows a plan view of the drilling pattern. The analysis considers a single
blasthole. This blasthole is at the intersection of the Y-axis and the X-axis. To
differentiate between this blasthole and the other blastholes, it is referred to as
the detonating blasthole throughout the text. The area of rock considered in the
analysis extends two times the burden on the sides and to the back of the blasthole,
and one burden in direction of the free face. This is the area inside the dashed
boundary in Figure 5.8. Because of the symmetry, only half the area need to be
analyzed. The hatchured section in Figure 5.8 is modeled. The dotted lines on the
figure contain the part of the burden which the blasthole is supposed to fracture and
displace when it is detonated. The geometry of the drilling pattern, the explosive
type, and the rock type are kept the same throughout this chapter.
Critical strain energy density is the failure criterion adopted to estimate the
potential fracture zone around the blast hole. This failure criterion was used by
Porter (1971, pp. 124-128), Porter and Fairhurst (1971), and Bhandari (1975, pp.
34-40, 146-147). The strain energy density, U, in plane stress, is calculated by
Where:
E=Young's modulus;
v=Poisson's ratio;
0'1 =major principal stress;
(5.10)
0'2=minor principal stress. The strain energy density associated with failure,
U" is given by
(5.11)
where O't is the ultimate tensile strength.
---- ._- .... _ .. -...
)(
BENCH FREE FACE
> • .),)," " • ), " " ,,".' < < < < < < « •
•
3m
, , , , , :E , ,(II) ,
•
-------------.-------------~
Figure 5,8 Layout of the drilling pattern showing the section modeled by the Finite Element Method. Hatchured section is the part modeled by the mesh. ~
c:.n ~
152
According to the strain energy density criterion, cracks can grow at any point
in the rock if U at this point exceeds U, (Porter, 1971, p.125). U, for Lithonia
granite is 232.706 N/m2•
In the finite element analyses, circular .boundaries are modeled by circular
edge elements, consistent loads are applied at the nodes, and 2 x 2 Gauss quadrature
order is used.
Figure 5.9 shows the mesh used to model the blast hole on the assumption
that no radial cracks exist around the blasthole or at the free face. All the elements
in this mesh are eight noded isoparametric elements. The mesh is composed of
143 elements and 486 nodal points. The boundary conditions along the symmetry
plane (X-axis is the trace of the symmetry plane) allow no displacement in the Y
direction. The borders of the mesh at the top (Y=6 m) and at the left (X= -6 m)
of Figure 5.9 are fixed to prevent displacements in X and Y directions. The free
face in front of the detonating blast hole (the right side, X= 3 m, of Figure 5.9) is
traction free. The applied internal pressure is 3.1717 x 109 N / m 2 (the upper bound
of the explosion pressure).
The problem is solved using the SABM program. Figure 5.10 shows the
displacement field. In the figure, the scale shows the maximum displacement, 1.5 x
10-2 m. All the displacements are normalized to that maximum displacement.
The plus signs are the locations of the Gauss points and the displacements at
these locations are represented by the arrows. To give a clearer picture of the
displacements, the displacements are magnified within some selected windows from
the domain. These windows are named A, B, C, and D shown in Figure 5.10 by the
dashed lines. The locations of the windows are selected so that they can be used
later to compare the displacement field with the displacement fields for the options
including radial cracks. Figure 5.11 shows the displacements within these windows.
Figures 5.10 and 5.11 show that the displacement field is symmetrical around the
blasthole within a radius about 0.3 times the burden. Outside this region, the effect
of the free face is seen by the increased displacements in front of the blast hole.
----, , --.. " ._--- •...
o G~I~----~------r------r------~----~---r--~r---~---T----~----~
~~~--~----~--+---~--+-~--~~~---+--~
e~J=~l-~---t--i--l--I-t-~-r==~_ -1&1
~
~~ J ) ~ ~ k: " l o 8 01 IN
>-
o c:i I ' I I I I 'J::IC:B:T:' , , i I I I iii 'I
-8.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 X-COORDINATE, m
Figure 5.9 Mesh used to model the blasthole without radial crackR.
sounding and core fracture logging with distance from the center of the blasthole.
Their results indicated that there are two zones of damage. Severe damage extends
to about 4 times the blasthole diameter. Less damage extends to about 7 blasthole
diameters. They gave a good summary of other measurements of the extent of
damage zones for some rock-explosive combinations. Extent of damage zone in
terms of charge diameter for some combinations are 9 - 10 for granite-C4 ; 21 - 27.5
for shale-60 percent dynamite; 7.5 - 11 for shale-ANFO; 10 - 15 for tuffaceous and
pyroclastic-60 percent dynamite; for soft rock 13 - 14.5, and in hard rock 10 - 11.5
(explosive is not reported).
From the above summary, the damage zone is larger for stronger explosive
in the same rock (in case of shale, using 60 percent dynamite produced damage 2
- 3 times the damage caused by ANFO). Hence, for Lithonia granite-semigelatine,
the extent of damage is supposed to be larger than 7 times the blasthole diameter
produced by Lithonia granite-ANFO. The damage limit in the back of the blasthole
is more or less associated with the extent of the widely spaced radial cracks. So
it can be considered the upper limit of radial crack extension. Another point is
that the explosive gas energy extends the radial cracks formed by the precursor
wave energy. This means that the length of the radial cracks to be used in the
quasi-static finite element analysis should be less than the extent of the damage
zone. In their theoretical analysis, Kutter and Fairhurst (1971) estimate the length
of radial cracks to be three times the hole diameter if no extension takes place due
to the reflected waves, 4 - 5.5 times the blasthole diameter if extension takes place.
They used the hole diameter and the crushed zone cavity as synonyms because
they assume that the crushed zone thickness is negligible. Using the crushed zone
diameter (28.4 cm for Lithonia granit-semigelatine), the crack length would be
---- . -_ ..... _ .. -...
169
8.5 times the blasthole diameter if extension due to reflected waves is ignored. If
extension due to wave reflection is considered, the crack length would be 11.3 - 12.7
times the blasthole diameter. The length of the radial cracks for the current study
is estimated to be 0.8 m (8 times the blasthole diameter and 2.82 times the crushed
zone diameter).
In the zone of widely spaced radial cracks, the number of radial cracks ranges
from 4 to 12. The number of these radial cracks from shock wave tests in a granite
disc is eight (Kutter and Fairhurst, 1971). So, the number of radial cracks in
Lithonia granite is assumed to be eight. It is understood that using this number
of radial cracks and/or their length may not be accurate. Ideally they should
be measured from field tests. However, they are considered to be in the correct
range. If field test data is available, the measured parameters for rock, explosive,
fractured zone boundary, length of radial cracks, number of radial cracks, fragment
size distribution can be used for the input parameters and for calibration of the
results of the model.
5.3.1 Non-pressurized Radial Cracks
Figure 5.15 shows the mesh used for modeling the blast hole when radial
cracks are included around the blasthole and at the free face. Crack tips are at the
tips of the heads of the arrows. To get better computation accuracy, Cook (1981,
p. 216) recommends that the length of the quarter point eight noded isoparametric
elements, along the crack surface, be less than roughly 0.3 times the crack length.
The length of the quarter point elements along the crack has been made much less
than 0.3 times the crack length. At each crack tip, four quarter point elements are
used to take care of the stress singularities at the crack tip. The circular boundaries
at the detonating blasthole and at the free face are taken at the boundaries of the
crushed zone at each blasthole location. The mesh is composed of 386 elements,
mostly eight noded isoparametric elements. A small number of transition elements
are used for transitions from smaller to larger size elements. Those transition ele
ments are five, six, or seven noded isoparametric elements. The mesh is composed
of a total of 1334 nodes.
I • ~, •. ~ " ... ~ .•.• _~ ___ ,_~_ •• ,,_ •
o .;
o C
e~ rzf ~ Zo - . Q'" £l: o 8 IC!
)401
C! -o o
-
-
-
-
-
, , -1.0 -1.0 -4.0
~~ ........... ~,
f"'oo.
~/~ ~ k-
~~
~ ~\ ~ t:: f"'oo.
I
\ J \
1j~ ~ I :;.. 1'0 I"'
- II J ~ J"l"-ll W"'I""lrrI , , I I
-3.0 -2.0 -1.0 0.0 1.0 2.0 :1.0 X-COORDINATE, m
Figure 5.15 Mesh used for modeling the blasthole with radial cracks at the blasthole and at the free face. The arrow heads represent the tips of the cracks.
.... -.,J o
171
The circular boundary of the crushed zone around the detonating blasthole
is loaded with an internal pressure equal to 50% of the detonating pressure. The
radial cracks are left unloaded. The free face is traction free. Displacements in the
Y -direction along the symmetry plane (Y = 0.0) are prevented except along the
surfaces of the cracks. The left (X = -6.0 m) and top (Y = 6.0 m) boundaries are
fixed except at the crack surface at the top where displacements are allowed in the
X-direction.
Figure 5.16 shows the displacement field. The maximum displacement in the
field is 4.28 X 10-2 m. Displacements toward and at the free face are much larger
than those in the back or at the side of the detonating blasthole. Figure 5.17 shows
the displacement field within some selected windows. These windows are defined
by the dashed lines in Figure 5.16. The dotted lines in Figure 5.17 represent the
radial cracks. It can be seen from Figure 5.17 that the radial cracks cause large
changes in the displacement field in terms of magnitude and direction. Most cracks
show unsymmetric displacements around their axes. The displacements are much
larger compared with those when radial cracks are not considered.
Comparing Figure 5.17 to Figure 5.11, we observe the following changes in
the displacements. In window A, the maximum displacement is increased from
3.4 7 x 10-4 m to 3.24 x 10-3 m, in window B from 1.36 x 10-3 m to 1.05 x 10-2
m, in window C increased from 1.97 x 10-3 m to 1.87 x 10-2 m, and in window D
increased from 1.5 x 10-2 m to 4.28 x 10-2 m. The increase in displacements close
to the blasthole is about 300%. At the free face the increase is more than 900%.
Figure 5.18 shows the stress field around the blasthole. The stress field
within some selected windows is shown in Figure 5.19. In Figure 5.18, we can see
some disturbance in the directions of the principal stresses at the free face and
some locations in which the state of stress is compressive. The picture is clearer
in Figure 5.19. Stress concentrations can be seen at the crack tips. Radial cracks
are represented by the dotted lines. Around the detonating blasthole and within a
region extending roughly to half the length of the radial cracks, the stress state is
almost uniaxial compression. Beyond this region, the tensile components begin to
appear. Along the free face, there are several locations where biaxial tension can
t t ,. tl- t .... .. : .. " t .. .. L.,.. ___ -.,. __
.. t t .. .. ..
* .. -# ., t .. .. ~
., • ~
t .. .. -# ., ",
",
t .. • of ./ ",
.. 6-...
~ .. • ott
-t. <tr .. ... ....
II • 4.20*10-e METERS
Figure 5.16 Displacement field around a blasthole when non-pressurized radial cracks are considered. ..... ~ ~
173
• •
J I I
/
3.24.10-1 METERS
Figure 5.17
(a) Displacement field within some selected windows when nonpressurized radial crades are considered. (a) Window A. (b) V\Tindow B. (c) V\Tindow C. (d) Window D.
I I I I ~"', ...... ~ I---.----~----.----. :'0 ~~ .".. : ,~ 11 ,.
I I I I I I I :, __ _~_1I : • • :. _____ =-_-.:=-_--=-_.:.1. --- - - ---- - - -- -
1.44-10' 2.07-10' PASCAL ••
Figure 5.18 Stress field around a bla.c;thole when non-pressurized radial cracks are considered. .... --l --l
I I f f
I
I +
f f f
t +
(a)
....
I
• III
• • , I ,
Figure 5.19 Stress field within some selected windows when non-~ressurized radial cracks are considered. (a) Window A. (b) 'Vindow B. (c) Window C. (d) Window D.
Figure 5.21 Displacement field around a blasthole when unifonnly pressurized radial cracks are considered. .... 00 tJ:>..
185
the displacement field. The displacement fields within selected windows are shown
in Figure 5.22. Figures 5.21 and 5.22 are compared with Figure 5.16 and Figure
5.17 to see the changes in the displacement field due to the uniform pressurization
of the radial cracks. Looking at the directions of the displacements, no significant
change can be seen outside the zone of radius extending to the tips of the cracks.
The displacement directions inside this zone change significantly. In Figure 5.22
(d), the displacements along the surfaces of the cracks show more diversion from
the directions of the cracks. This diversion favors increasing the openings of the
radial cracks. Compare displacements in figure 5.22 (d) to those in Figure 5.17 (d).
At the free face in window A the maximum displacement has increased from
3.24 X 10-3 m to 3.08 x 10-2 m, in window B increased from 1.05 x 10-2 m to
1.00 x 10-1 m, in window C increased from 1.87 x 10-2 m to 1.78 X 10-1 m, in
window D increased from 4.28 x 10-2 m to 2.81 x 10-1 m. Close to the detonating
blasthole, the displacements increase seven times by the uniform pressurization of
the cracks. Close to the free face, the displacements increase roughly 10 times.
Figure 5.23 shows the stress field. The stress fields in some windows are
shown in Figure 5.24. A region of biaxial compressive stress forms around the det
onating blasthole. This region extends out up to the tips of the cracks. Outside
this region, the sense or general distribution of the stresses does not change signifi
cantly. The stresses increase. Comparison between Figure 5.24 and Figure 5.19 (for
non-pressurized cracks) shows these increases. In window A, the maximum stress
has increased from 1.16 X 108 to 1.10 X 109 pascal, in window B from 1.02 x 108 to
9.72 X 108 pascal, in window C from 3.80 x 108 to 3.61 X 109 pascal, and in window
D it has increased from 1.44 x 109 to 9.65 X 109 pascal. Close to the detonating
blasthole stresses increase about seven times and close to the free face the stresses
increase about ten times due to the uniform pressurization of the radial cracks.
Figure 5.25 shows the contour map for the scaled strain energy density. Com
paring Figure 5.25 with Figure 5.20 (for non-pressurized cracks), the general shape
of the contours is similar except at the free face where the contours are pushed closer
to the face when the cracks are pressurized. The contour levels have increased by
about 100 times their values in Figure 5.20. Compare contour levels of 50000,5000,
186
•
.t J
I I
/ /
(a)
Figure 5.22 Displacement field within some selected windows when uniformly pressurized radial cracks are considered. (a) Window A. (b) Window B. (c) Window C. (d) Window D.
Figure 5.23 Stress field around a blasthole when uniformly pressurized radial cracks are considered. co 0
I I f
f I / +
f f f
1
• III
• • , I
191
Figure 5.24 Stress field within some selected windows when uniformlr pressurized radial cracks are considered. (a) \Vindow A. (b) Window B. (c) Window C. (d) Window D.
Figure 5.26 Displacement field around a blasthole when linearly pressurized radial cracks are considered.
.... to -.J
198
.. • .. " ... .. ~
./1-1 4
4 ., 4 ./ I
.t
J I I
/ /
METERS
(a)
Figure 5.27 Displacement field within some selected windows when linearly pressurized radial cracks are considered. (a) Window A. (b) vVindow B. (c) \\Tindow C. (d) Window D.
Figure 5.28 Stress field around a blasthole when linearly pressurized radial cracks are considered. ~ 0 Col
I f f f I I +
f f
(a)
....
• • • I
• , I
Figure 5.29 Stress field within some selected windows when linearly pressurized radial cracks are considered. (a) Window A. (b) Window B. (c) Window C. (d) Window D.
Figure 5.30 Contour map for the scaled strain energy density around the blast hole when linearly pressurized radial cracks are considered.
~ o 00
209
pressurized or non-pressurized cracks' system would be the same. In this later
case, crack pressurization should be associated with decrease in the gas pressure
within the cavity and the cracks to satisfy the constant energy input condition.
Development of measurement techniques to provide this information can bring more
solid understanding of the real timing and extent of crack pressurization.
5.4 Blasthole Equivalent Cavity and Radial Cracks
The equivalent cavity is a circular cavity of diameter equal to or greater than
the nominal blasthole diameter. It is introduced to simplify modeling of the blast
hole. It is assumed that if the internal gas pressure is applied along the boundary of
the equivalent cavity, the stress field produced outside the cavity is approximately
the same as the stress field produced by including the details of the complex non
linear nonelastic behaviour of the zone close to the nominal blast hole boundary. It
has been shown in the beginning of this chapter that there is some ambiguity in
the use of the equivalent cavity to model bench blasting. Some researchers use the
nominal blasthole, ignoring the existence of the nonlinear zone. Others use a cavity
diameter equal to the crushed zone diameter, equal to the diameter of the zone of
the radial cracks, or some diameter in between these two.
Kutter (1967) proved that the boundary of pressurized radial cracks can be
replaced by an equivalent cavity of boundary equal to the boundary of the tips of
the radial cracks. His mathematical solution is for infinite plate and plane strain
conditions. Representation of radial cracks in the mesh for finite element analysis
complicates the mesh and node numbering. In addition, it increases the demand
for storage memory and for computation time because of the greater number of
elements and larger band width required.
In this section, the validity is investigated of using an equivalent cavity equal
to the boundary of the tips of the radial cracks in modeling bench blasting. In the
mesh of Figure 5.15, the elements around the blasthole up to the tips of the radial
cracks are removed. An internal pressure of magnitude equal to that applied at the
boundary of the crushed zone is applied to the new boundary. Figure 5.31 shows
o .;
o ~
e~ rai ~ Zo - . Q'" 0: o o tJ It:!
)oIW
t:! -o o
-
-
-
-
-
I , , -e.o -1.0 -4.0
~~ "- ..) ~\
f""oo. r-
~~ -,~~ ~ ~
~~ ~ f""oo. r-
\ 1 I
~ ~ ~ I ~
~11 ~ ~
n T 1 ~ , I • I
-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 X-COORDINATE. m
Figure 5.31 Mesh used to model the blasthole using equivalent cavity equal to the zone of radial cracks. t-.) J-A o
211
the mesh after the removal of the zone of radial cracks. The mesh is left with 326
elements and 1114 nodal points.
The general displacement field is shown in Figure 5.32 and the displacement
fields within the selected windows are shown in Figure 5.33. Comparison between
the displacement fields produced by the e~uivalent cavity and the uniformly pres
surized crack condition is possible now. Figures 5.32 and 5.33 show displacement
directions similar to those in Figures 5.21 and 5.22. The displacements produced by
the equivalent cavity are much smaller than those produced by including the pres
surized radial cracks. In window A, the maximum displacement is reduced from
3.08 x 10-2 m to 9.78 X 10-3 m when radial cracks are replaced by an equivalent
cavity. In window B, the maximum displacement is reduced from 1.00 x 10-1 m to
3.23 x 10-2 mj in window C, it is reduced from 1.78 x 10-1 m to 5.75 x 10-2 mj in
window D, it is reduced from 2.81 x 10-1 m to 6.52 x 10-2 m. On the average, the
displacements are reduced to one third their magnitudes close to the free face and
to one fourth their magnitudes close to the detonating blasthole.
Figure 5.34 shows the general stress field and Figure 5.35 shows the stress
fields within the selected windows. Comparing Figures 5.34 and 5.35 to Figures
5.23 and 5.24 (for uniformly pressurized radial cracks), we can see similar stress
fields outside the equivalent cavity in both cases from the point of view of type and
direction. However, the stress magnitudes are reduced drastically by replacing the
radial cracks by an equivalent cavity. In window A, the maximum stress is reduced
from 1.10 x 109 to 3.49 X 108 pascal, in window B from 9.72 x 108 to 3.14 X 108
pascal, in window C from 3.61 x 109 to 1.17 X 109 pascal, and in window D from
9.65 x 109 to 9.69 X 108 pascal. Roughly speaking, the stresses close to the the
equivalent cavity are decreased to one tenth their magnitudes and the stresses close
to the free face are decreased to one third their magnitudes when radial cracks are
replaced by the equivalent cavity. . Figure 5.36 shows the contour map for the scaled strain energy density. Com-
paring Figure 5.36 with Figure 5.25 where the pressurized radial cracks are consid
ered, we observe that the general s3ape of the contours is similar. The contour
:A .. t .. : ...
• • t t .. .. ~., of. .. : .,1 .. .. ., L~---4--
--./
t .. .. -I ., ,L ___ .L __ ., ., ~B J' ..
/ . .1
* • ., '" I'
.f.
t ... ., ~ / .." .I .. .. .. 11 ..
"" \---... .. "-.. 'i ~ ... • ~ .. .....
6.52-10-· METERS
Figure 5.32 Displacement field around a blasthole when the zone of radial cracks is replaced by an equivalent cavity.
t-) -t-)
213
• ;#
.t J
I I
/ / 1 / I / / I
/ ~ (a)
Figure 5.33 Displacement field within some selected windows when the zone of radial cracks is replaced by an equivalent cavity. (a) Window A. (b) Window B. {c) Window C. (d) Window D.
Figure 5.34 Stress field around a blasthole when the zone of radial cracks is replaced by an equivalent cavity. '" ~ -.J
I I f
I
I
/ +
f f f
(a)
I
• • • • , I ,
218
Figure 5.35 Stress field within some selected windows when the zone of radial cracks is replaced by an equivalent cavitr. (aj Window A. (b) Window B. (c) Windo'v C. (d) Window D.
TENSILE STRENGTH (IN TERMS OF STATIC TENSILE' STRENGTH)
Figure 5.41 Variation of the normalized areas of the scaled strain energy density contours with tensile strength when radial cracks are not considered.
234
Contour levels 100 and 1000 also show high decreasing rate when the tensile strength
is increased from 1 to 2 times the static tensile strength. Then their nonnalized
areas show continuous decrease at a decreasing rate with increasing tensile strength.
The normalized areas of the contours show less sensitivity to the increase of tensile
strength as the contour levels increase. With increasing tensile strength, the contour
areas shrink and contours of level 1 or more are disconnected from the free face and
become symmetric around the blasthole. Figure 5.42 shows the contour map of
the scaled strain energy density when tensile strength is 6 times the static tensile
strength. Compare Figure 5.42 with Figure 5.14 where the tensile strength equals
the static tensile strength.
Figure 5.43 shows the variation of the nonnalized contour areas of the scaled
strain energy density with tensile strength when non-pressurized radial cracks are
considered. The normalized contour areas decrease at a decreasing rate as the
tensile strength increases. As in the case of no radial cracks, the contour areas
are more sensitive to increasing the tensile strength from 1 to 2 times the static
tensile strength than to further increases of the tensile strength. In this range,
the normalized contour areas show a steep decrease. Including the non-pressurized
radial cracks, has increased the nonnalized areas of the contours by about 200% of
the normalized contour areas when radial cracks are ignored.
Figure 5.44 shows the variation of the nonnalized areas of the scaled strain
energy density with tensile strength when unifonnly pressurized radial cracks are
considered. Contours 1 and 10 do not show much variation in their areas at low
magnitudes of tensile strength. This not because of the unsensitivity of the contours
to the tensile strength but rather because the two contours are almost displaced out
of the map by higher contour levels. This can be seen in Figure 5.25. If the mesh
size were larger, they would have shown the same behaviour as contour levels 100
and 1000. Uniformly pressurizing the radial cracks has increased the nonnalized
areas of the contours by 400 - 600% of the contour areas when radial cracks are not
pressurized. Contours 100 and 1000 show sharply decreasing rates when the tensile
strength is increased from 1 to 2 times the static tensile strength. When the tensile
strength is increased more, the decrease of the normalized contour areas continues,
1.0 1.0 '.0 •• 0 0.0 '.0 7.0 '.0 '.0 10.0 TENSILE STRENGTH (IN TERMS OF STATIC TENSILE STRENGTH)
Figure 5.43 Variation of the normalized areas of the scaled strain energy density contours with tensile strength when non-pressurized radial cracks are considered.
TENSILE STRENGTH (IN TERMS OF STATIC TENSILE STRENGTH)
Figure 5.44 Variation of the normalized areas of the scaled strain energy density contours with tensile strength when uniformly pressurized radial cracks are considered.
238
but at a decreasing rate.
Figure 5.45 shows the variation of the normalized areas of the contours with
tensile strength when the radial cracks are linearly pressurized. The behaviour of
the normalized contour areas here is the same as in the case of uniformly pressurized
cracks. The normalized areas of the contours are less than for the case of uniformly
pressurized radial cracks and greater than for the case of non-pressurized radial
cracks.
In all cases, the effect of increasing the tensile strength on the shape of the
contours is more or less moving the contours to the shape and location of a higher
contour level.
In chapter four, it has been estimated that the dynamic compressive strength
of Lithonia granite is 9 times its static compressive strength. Accordingly, the dy
namic tensile strength can be estimated to be 9 times the static tensile strength.
Out of all the contour maps for the scaled strain energy density, the ones for tensile
strength equal to 9 times the static tensile strength are presented here. Figures
5.46 through 5.49 show the contour maps for the scaled strain energy density for
tensile strength equal to the postulated dynamic tensile strength (i.e. 9 times the
static tensile strength). These figures are for models considering no-radial cracks,
non-pressurized radial cracks, uniformly pressurized radial cracks, and linearly pres
surized radial cracks respectively. The internal pressure, Pi, is 50% Pd.
From Figure 5.46, we can see that when radial cracks are not considered,
contour level 1 contains almost a circular region around the blasthole. The radius
of this region is less than half the burden. Keeping in mind that the upper limit of
internal pressure is applied, modeling the blast hole without radial cracks does not
give an acceptable estimation of the fractured zone.
Figure 5.47 (for non-pressurized radial cracks) shows a better shape for the
fractured zone. The fracture zone extends toward the free face to distances beyond
the tips of the radial cracks of the previously blasted holes. This means that the
fracture zone extends through the free face and has some widening at the free face.
The figure also shows some isolated fracturing at the free face at the tips of cracks
extending from the previously blasted holes at distances greater than the burden
1.0 1.0 1.0 f.O 1.0 '.0 ?O 1.0 1.0 10.0 TENSILE STRENGTH (IN TERMS OF STATIC TENSILE STRENGTH)
Figure 5.45 Variation of the normalized areas of the scaled strain energy density contours with tensile strength when linearly pressurized radial cracks are considered.
Figure 5.46 Scaled strain energy density contour map when radial cracks are not considered using the dynamic tensile strength and internal pressure 50% of the detonation pressure. ~
o
6. i i iilll KI
5
.. E w ~ Z 3 0 ~ 0 0 (J
.J- 2~
t'
°_6
Figure 5.47
~ ///
r~ . .. ~ ,0
-5 -4 -2 -t 2 X -COORDINATE. m
Scaled strain energy density contour map when non-pressurized radial cracks are considered using the dynamic tensile strength and intemal pressure 50% of the detonation pressure.
t-) M:>o ......
E w ~ z o c::: a a u I
r
1
~
o o
-5
r;:~~ !'>O 'OO~100
It) ~
-4 -2 -t X-COORDINATE. m
Figure 5.48 Scaled strain energy density contour map when uniformly pressurized radial cracks are considered using the dynamic tensile strength and internal pressure 50% of the detonation pressure.
I'-' ~ tv
E w ~ z 0 0:: 0 0 (J
I >- 2
I I I
-; 0 ~ . -
1 rl/: ~ 0<:5
"
°_6 -5 -3 -2 -t X-COORDINATE, m
Figure 5.49 Scaled strain energy density contour map when linearly pressurized radial cracks are considered using the dynamic tensile strength and internal pressure 50% of the detonation pressure. ~
Col
244
from the symmetry plane. Figures 5.48 and 5.49 (for pressurized radial cracks) show
very large fractured areas. These predicted zones are unacceptable. However, they
are promising because when the applied internal pressure is decreased, they may
give good predictions.
The decrease of the normalized contour areas with increasing ten'3ile strength,
from 1 to 10 times the static tensile strength, at a decreasing rate; the sharper de
crease of the normalized contour areas when the tensile strength is increased from 1
to 3 times the static tensile strength; the smaller sensitivity of higher contour levels
to an increase in tensile strength; all these variations can be considered equivalent to
increasing the ratio of (J't!(J'c from 1.5% to 15% or the ratio of (J't-dYRamic/(J't-atatic
from 1 to 10 within the range shown in Table 5.1. This means that for a given explo
sive, the fractured area is smaller for rocks of higher (J't! (J'c and (J't-dYRamic/ (J't-atatic
ratios.
5.6 Effect of the Explosion Pressure on the Strain Energy
Around the Blasthole
Because of the catastrophic and short time nature of the detonation process,
the very high pressures and temperatures associated with it makes measurement
of the pressure-time history in a blasthole very difficult, if not impossible. Ex
plosives' manufacturers, based on thermohydrodynamic and thermochemical calcu
lations, have developed empirical relationships for approximate estimations of the
detonation pressure (e.g. equation 4.2 and equation 4.3). As mentioned earlier in
this chapter, the upper limit for the gas pressure resulting from a detonation should
not exceed 50% of the detonation pressure (Hino, 1959, p. 61). Because of the
crushing of the rock immediately around the charge, the widening of the hole due
to the inelastic deformation outside the crushed zone, and the unideal detonation
due to field conditions, the gas pressure drops to values less than this 50% of the
detonation pressure.
Laboratory measurements of detonation pressure for detonations in steel
pipes showed that measured pressures are less than theoretical pressures (Barnhard
245
and Bahr, 1981). These data show high dependency of the accuracy of the calcu
lated pressure on the diameter. They showed that 94/6 ANFO develops 80% of
its theoretical pressure at 8 inch diameter and 70% at 6 inch diameter. A 10%
aluminized ANFO develops 75% of theoretical pressure in a 6 inch diameter hole.
Packaged COMSOL blasting agents exceeded 50% of the theoretical pressure in 3
inch diameter hole.
The explosion gas pressures are much higher than the static compressive
strength of rocks. At the boundary of the crushed zone, the gas pressures drop to
magnitudes equal to the dynamic compressive strengths of rocks. Because dynamic
strengths of rocks are higher than their static strengths, the lower bound of the
dynamic strength is the static strength. Hence, it is reasonable to assume the lower
bound of the gas pressure to be equal to the static compressive strength of the rock.
In this section, the internal gas pressure effect on the scaled strain energy density
distribution around the blasthole, is studied within the upper (50% Pd ) and the
lower limit (static compressive strength) of the gas pressure. The internal pressure,
Pi, is varied by multiplying the static compressive strength of Lithonia granite by
a factor, n. Table 5.2 shows the range of the internal blasthole pressures used.
Table 5.2 Range of Borehole Pressures Applied in the Models. Factor n is multiplied by u c to determine the borehole pressure.
n Pi = n x uc, 109 N/m2 (PdPd) x 100
1 0.206850 3.26
3 0.620550 9.78
5 1.034250 16.30
7 1.446985 22.83
9 t1!~1650 29.35
11 2.275350 35.87
13 2.689050 42.39
15.33 3.171700 50.00
246
The tensile strength used here is the Lithonia granite dynamic tensile
strength (9 times the static tensile strength). For each applied internal pressure,
Pi, the contour map for the scaled strain energy density is constructed. The areas
inside contour levels 1, 10, 100, and 1000 are measured for each contour map using
a planimeter. These areas are normalized by dividing them by 1/2(burden)2.
Figure 5.50 shows the variation of the normalized areas of the scaled strain
energy density contours with increasing internal pressure when non-pressurized ra
dial cracks are considered. The contour areas increase with increasing internal
pressure. The rate of increase of the normalized areas of the contours decreases
with increase in the contour levels. Contour level 1 shows the highest rate of in
crease. Increasing the internal pressure from 3.26% Pd (1 O'c) to 50% Pd (15.33 O'c)
has increased the normalized area of contour level 1 from 0.09 to 3. In other words,
increasing the internal pressure 15.3 fold, causes the fractured zone to increase 33.3
fold. So, the rate of increase of the fractured zone is more than twice the rate of
increase of the internal pressure. Contour levels 100 and 1000 show small rates
of increase. This means that using more powerful explosives adds more to the
fragmentation than to the highly fractured zone.
Figure 5.51 shows the contour map for the model of the non-pressurized
radial cracks when internal pressure is 9 times the static compressive strength (i.e.
Pi = dynamic O'c). So, the figure shows the contours of the scaled strain energy
density when both compressive strength and tensile strength are dynamic. Contour
level 1 goes beyond the tips of the radial cracks close to the symmetry plane. This
means that fracturing can take place up to the free face. However, the shape of
the fractured area narrows at the free face instead of widening up. This can be
interpreted in two ways. Either the non-pressurized cracks condition is not a good
idealization of the fracturing process and some crack pressurization has to take
place, or the burden analyzed is larger than the optimum burden for this explosive
rock combination.
Figure 5.52 shows the variation of the normalized areas of the contours of
the scaled strain energy density when uniformly pressurized cracks are considered.
• Ii
. ' " " " ,m •• , "
.' " " .' .a ..... .... " "
0' .. , .' .' .' .' .' .a ....
.' .' .' .' .'
247
.' ·.0
....... ~ ...... -6
~ a.·--/ ,-..... """ .... .. .... "
O •••••• d .....A. ...... -6-.- ... -....-,; ...... .-- .. ----
'.0 '.0 10.0 IG.O 10.0 ID.O 10.0 IG.O 40.0 40.0 GO.O INTERNAL PRESSURE (AS PERCENTAGE OF THE DETONATION PRESSURE)
Figure 5,50 Variation of the normalized areas of the scaled strain energy density with the internal pressure when non-pressurized radial cracks are considered using the dynamic tensile strength,
----- -_.,.
6. \ it II \<:1
6
4 E w ~ Z 3 o 0:: o o u , r 2
1
°_6
Figure 5.51
-5 -4 -2 -1
X-COORDINATE. m
Contour map of the scaled strain energy density for internal pressure equal to the dynamic compressive strength when non-pressurized radial cracks are considered.
0.0 1.0 10.0 18.0 10.0 18.0 10.0 18.0 40.0 48.0 80.0 INTERNAL PRESSURE (AS PERCENTAGE OF THE DETONATION PRESSURE)
Figure 5.52 Variation of the normalized areas of the scaled strain energy density with the internal pressure when uniformly pressurized radial cracks are considered using the dynamic tensile strength.
250
The contour areas increase with increase in the internal pressure. The rate of in
crease of the contour areas decreases as the contour level increases. Also the rate of
increase decreases with increase in the internal pressure. This behaviour is similar
to that in Figure 5.50 for non-pressurized radial cracks. Uniformly pressurizing the
radial cracks, has substantially increased the areas of the contours. This increase is
about 100 times their magnitude compared to non-pressurized radial cracks (com
pare Figure 5.52 to Figure 5.50). The normalized areas of contour levels 1 and 10
show a steeper decrease in their slopes than contour levels 100 and 1000 when the
internal pressure is increased to 20% Pd. This is because contour levels 1 and 10
extend beyond the dimensions of the mesh, and hence the increase in their areas as
presented in the figure is less than they should be.
Figure 5.53 shows the contour map of the scaled strain energy density for
an internal pressure equal to the dynamic compressive strength of the rock when
uniformly pressurized radial cracks are analyzed. The fractured area produced by
uniformly pressurized radial cracks is extremely large. Accordingly, the idealization
option of uniform pressure applied to the total length of the radial cracks may be
rejected.
Figure 5.54 shows the variation of the normalized areas of the scaled strain
energy density contours with increase in internal pressure when the radial cracks
are linearly pressurized. The figure shows the same trends as in the non-pressurized
and uniformly pressurized radial cracks. The normalized areas of the contours
increase with increase in the internal pressure. The areas of higher level contours
show smaller rates of increase than the areas of lower level contours. The normalized
areas of the contours are less than for the case of uniformly pressurized radial cracks
but much larger than for the non-pressurized radial cracks.
Figure 5.55 shows the contour map for the scaled strajn energy density for
internal pressure equal to the dynamic compressive strength when linearly pressur
ized radial cracks are considered. The produced fractured area is still large. So, the
option of linearly pressurizing the total length of radial cracks may also be rejected.
&. ii Ii It I'Ia:I
E . W
~ Z o ~ o o U I
>- 2
1
Figure 5.53
~ ~ 8 o o· •. : Cr)
-5 -4 -3 -2 -1 X-COORDINATE, m
Contour map of the scaled strain energy density for internal pressure equal to the dynamic compressive strength when unifonnly pressurized radial cracks are considered.
0.0 '.0 10.0 10.0 10.0 10.0 10.0 10.0 40.0 40.0 10.0 INTERNAL PRESSURE (AS PERCENTAGE OF THE DETONATION PRESSURE)
Figure 5.54 Variation of the nonnalized areas of the scaled strain energy density with the internal pressure when linearly pressurized radial cracks are considered using the dynamic tensile strength.
Contour map of the scaled strain energy density for internal pressure equal to the dynamic compressive strength when linearly pressurized radial cracks are considered.
t>:) 01 c,.)
254
According to the trends of the above analyses, pressurizing a small length of
the radial cracks can bring the shape of contour level 1 to more compatible shape
with the real fractured zone shape.
Assuming a constant dynamic tensile strength, the above behaviour of the
normalized area of the contours of the scaled strain energy density with incr~asing
internal pressure can be extended to explain the variation of the normalized con
tour areas in terms of increasing compressive strength and in terms of increasing
the ratio (1c/(1t. In other words, the normalized areas of the contours of the strain
energy density increase at a decreasing rate with increase in the dynamic com
pressive strength if the tensile strength is considered constant. The increase rate
of the normalized areas is higher for lower contour levels than for higher contour
levels. The normalized areas of the contours of the strain energy density increase
at a decreasing rate with increase in the (1c/(1t rat.io and the increase rate of the
normalized areas is higher for lower contour levels than for higher contour levels.
This behaviour correlates well with what has been seen before that the normalized
areas of the contours decrease with increase in the tensile strength for a constant
internal borehole pressure.
5.7 Summary
Modeling of a blasthole using two dimensional quasi-static finite element
analysis is investigated from both the computational and idealization points of view.
Modeling of a circular hole subjected to internal pressure is investigated. More
accurate stress calculations are obtained when the circular boundary is modeled by
circular side elements and when consistent loads are applied at the nodal points.
The radial length of the elements of the first layer around the hole is taken equal
to half the hole radius. When the boundary of the hole is modeled by 12 elements,
the errors in radial stresses, due to the combined use of straight line segments and
lumped loads, are about 10% at a distance from the hole boundary of about 20% of
the hole radius. The straight line representation of the boundary and the lumped
loads contribute approximately equally to the error. The tangential stresses show
Table B.7 Sample of Fitting Equations Obtained by the Curve Fitting Program for the Relation between the Scaled Crushed Zone Diameter and the Velocity Ratio. Y is the scaled crushed zone diameter and X is the velocity ratio.
Square of the Correlation Factor (R2) .
Equation Salt Granite Limestone (3 inch)
Y = l/(A * (X + B)2 + C) 0.9176 0.7267 0.2372
Y = X/(A * X + B) 0.8827 0.7077 0.2024
Y = A *X(BIX) 0.8997 0.6858 0.2169
Y=A*Bx *Xo 0.9029 0.6835 0.2333
Y = A * e«X -B)/2) 0.9041 0.6773 0.2321 Y = A. * e((lnX _B)2 10) 0.9025 0.6849 0.2338
Table B.8 Sample of Fitting Equations Obtained by the Curve Fitting Program for the Relation between the Scaled Crushed Zone Diameter and the Characteristic Impedance Ratio. Y is the scaled crushed zone diameter and X is the characteristic impedance ratio.
Square of the Correlation Factor (R2)
Equation Salt Granite Limestone (3 inch)
Y = X/(A *X +B) 0.9140 0.6684 0.2989
Y = l/(A * (X + B)2 + C) 0.9318 0.6248 0.3173
Y = l/(A + B * X) 0.5824 0.5385 0.3152 Y = A * e((lnX _B)2 10) 0.9148 0.6338 0.3053
Y = A * e«X-B)/2) 0.9203 0.6107 0.3060
Y = A * B(l/X) * XO 0.9131 0.6367 0.3079
297
298
Table B.9 Sample of Fitting Equations Obtained by the Curve Fitting . Program for the Relation between the Scaled Crushed Zone Diameter and the Medium Stress Ratio. Y is the scaled crushed zone diameter and X is the medium stress ratio.
Square of the Correlation Factor e R2)
Equation Salt Granite
Y=X/(A*X+B) 0.9180 0.7013
Y = l/(A * eX -+- B)2 + C) 0.9241 0.6591
Y = A * XB * (1- X)c 0.9193 0.6252
Y = A * e((lnX _B)2 /C) 0.9064 0.6540
Y=A*B{l/X)*Xc 0.9055 0.6570
Y = A * e«X-B)/2) 0.9111 0.6234
Table B.10 Sample of Fitting Equations Obtained by the Curve Fitting Program for the Relation between the Scaled Crushed Zone Diameter and the Detonation Pressure Ratio. Y is the scaled crushed zone diameter and X is the detonation pressure ratio.
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