A New Technique for Predicting Rock Fragmentation in Blasting
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7/27/2019 A New Technique for Predicting Rock Fragmentation in Blasting
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A New Technique for Predicting Rock Fragmentation Blasting
P A Persson
STR CT
The explosion
of
a charge in a drillhole sets the surrounding rock mass
into vibrating stress wave motion. Except
the immediate vicinity
of
the
drillhole, the dynamic stresses associated with this motion do damage
only to pre-existing joints, cracks, or other weak planes, not to the rock
material in between these. The joints are weak
in
tension, therefore the
damage occurs
as
a result
of
tensile stresses. The initial damage process
in the rock mass that ultimately breaks loose in front
of
the hole is similar
to that in the remaining rock behind the drillhole. Recorded or calculated
values
of
the v ibr ation velocity and frequency contain a wealth
of
information about the combination
of
stress and strain that causes the
damage.
This paper outlines a new technique by which the peak strain energy
derived from measured or calculated vibration velocity records
is
used to
de te rm in e the local fragment s ize distribution. combines two
previously known and well tested techniques, namely the
Holmberg-Persson calculation
of
the
peak
vibration velocity generated by
an extended charge and King s calculations
of
the fragment size
distribution
as
a function
of
the strain energy
in
rock crushing. Both of
these calculations are based on experimental data and have been tested
and found
to
agree well with actual conditions in their respective fields.
Holmberg-Persson s calculated peak vibration velocities have been used
successfully to predict and control damage to the remaining rock in
cautious blasting, while King s calculation successfully describes the
comminution
of
rock in mechanical crushing.
Preliminary predictions
of
fragmentation in two types
of
rock blasting,
a large hole open pit mining blast and a tunnel round, indicate that the
new technique for fragmentation prediction has the potential for
predicting fragment size distributions within the rock removed by the
blast.
Two types
of
experiments are proposed to further evaluate the strain
energy concept for predicting rock damage and fragmentation in blasting.
STR INENERGYVERSUS VffiR TION
VELOCITY
The high pressure
of
the detonation reaction product gases acting
on the drillhole wall gives rise to a shock wave in the surrounding
rock. For a drillhole fully loaded with a high-energy high-density
explosive, the combined stress es behind the shock w ave front
may exceed the strength of the rock material, causing large plastic
deformation and crushing of the rock material near the drillhole.
As the drillhole expands, the compressive stresses are relieved,
and therefore, only a small region around the drillhole is exposed
to this large plastic deformation and crushing. Depending on the
strength
of
the rock material and the energy density of the
e xpl os iv e, this region e xte nds no further than about o ne hole
diameter outside of the drillhole wall. For holes in hard rock
loaded as required for smooth-blasting or pre-splitting, no plastic
deformation or crushing occur s at all. This is evidenced by the
half- dr illholes r emaining on the r ock face
of
a well designed
smooth-blast or pre-split.
The rest of the rock around a blasthole is exposed
to
combined
peak compressive stresses which are below the dynamic elastic
limit of strength of the homogeneous rock material. However,
after the peak compressive stresses have decayed, tensile stresses
occur, which may cause fracture of joints and widening of
pre-existing cracks. To under stand and be able to calculate the
extent of tensile stress damage to the joint structure of the rock
I. Director, Research Center for Energetic Materials, New Mexico
Institute
of
Mining and Technology, Socorro, NM 87801, USA.
mass, we need
to
know what combination
of
stresses and strains
cause damage to which joints.
Strain energy is used extensively as an intensity variable in
rock crushing and comminution. Vibration particle velocity and
frequency are similarly used as intensity variables in predicting
rock and building damage caused by ground vibrations from rock
blasting. However, in a vibrating rock mass a given peak particle
velocity and its related frequency also define and can be
translated into) a peak strain energy. The purpose of the work to
be described in the following was to investigate if the wealth of
information gathered about rock break-up in cr ushin g ca n b e
applied to fragmentation by blasting.
Consider, to clarify the concepts, a flat rock surface and an
explosive charge that sets up wave motion in the rock below and
at that surface as schematically shown in Figure
The compressive wave, in seismology called the P-wave, has
the highest velocity,
p
Though transmitting a high stress, it is
of
very short duration, therefore t he material mot ion in it is
negligible. The P -wave is f ollowed by a s hear wave, called the
S-wave, which propagates at a lower velocity
s
also with little
material motion. The major motion at the surface is that due to
the Rayleigh wave, the R-wave, a surface wave resulting from the
relaxation of shear stress, which propagates with the still lower
velocity
R
The shear wave originates at the surface at the front
of the compressive wave and sets up a shear stress in the material
behind it. It is the relaxation of this shear stress that gives rise to
the Rayleigh wave). If we draw an instantaneous cross-section of
the ground surface in the region where the Rayleigh wave is, the
surfac e will be wavy, as sh own with an e xagge rat ed vertical
amplitude in Figure
2
Typical values
of
the three wave velocities
in hard rock are
=
5000 m/s,
CS =
3500 m/s, and
R=
3000
m/s.
The ground sur face as indicated in F igur e 2, is bent, in a wavy
fashion, and consequently, the material is in a state of stress,
which varies periodically. The highest compressive stress is at
the bottom of the deepes t trough, the highest tensile stress is at
the crest
of
the highest wave. Where the surface is at its original
location, an inflection point) there is no stress.
The
actual shape
of the wavy surface is determined by the vertical particle velocity
and the constant) wave velocity
R
The strain which is the
FIG I - Far field stress waves at and below a flat rock surface
ground vibrations). P denotes the compressive stress wave, S the shear
wave, PS the shear wave originating at the surface, and R the
Rayleigh surface wave.
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P-A PERSSON
This stress is about two-thirds of t he t en si le str en gt h of
h om og en eou s gr ani te . H ol mb er g and Per sson c on cl ud ed from
their studies
of
bl ast d am ag e to a g ra ni te rock m ass t ha t t he peak
particle velocity
I
rnIs corresponding to this stress was the limit
w he re da mage in the form of opening of j oi nt s w oul d be gi n to
occur.
F rom Equati on 2 w e find the s trai n e ne rgy c or res pon di ng to
this stress level
11*60000
10 3
e
s
E= - =
= 0.00333 MJ m =
1 33J kg
2 c
2
2 3000
2
3000
for a rock material of density 2500 kglm
3
.
I
STRAIN ENERGY RELATED
TO
FRAGMENTATION
o r . : : : . . _ o : : : : ~ . . . . L . . _ . . . . . . d : = : : : : : L - - - - O J
o 0.2 0.4 0.6 0.8 1
Size Relative to Parent Particle
=
1
-lOO
-1000
,
-10,000
80
l
c:
60
I
n
II
l
40
II
20
Fla 3 -
Diagram
of
fragment
size distributions showing the
mass
fraction
of a crushed
rock
boulder passing a sieve
as
a function of the sieve
mesh
width
expressed
as
a fraction of the original
rock
boulder size), with the
strain energy applied
in
crushing
as
a parameter Lownds,
1995
M il in 199 4) c arr ie d o ut an e xt en si ve stu dy a nd a na ly si s
of
the
fragment size in comminution
of
rock by crushing. Using the
strain energy as a parameter, he was able to find agreement
between experimental and theoretical fragment size distributions
of rock crus hed by mechanical means. F igure 3 s how s M il in s
results as adapted by Lownds 1995), in the form of sieve
analysi s curves, s ho wi ng the mass fraction of the fragments
obtai ned from cr us hi ng o ne s ingl e pi ece of rock. The mass
fraction is plotted as a function of the s ieve openi ng size,
expressed as a fraction of the original size of the original piece
of
rock. T he or igi na l r oc k pi ece s w er e r ep re se nt ed b y a c ur ve wit h
t he strai n e ne rgy I J/k g. E xp er im en ta l sie ve c ur ve s for c ru sh ed
ro ck c ove re d t he r ange of strain energie s from 10000 J /kg to
10 J/kg, with corresponding values of the 50 per ce nt passing
sieve size ranging from 0.25 to 0.99.
100
r - - - - . - - - r - - - - - - - := - - - - . . . . - i i
I
Energy
J/kg
0
1
-10
We can now compare the lower limit strain energy for incipient
rock damage derived from Holmberg and Persson s
me asur eme nt s i n l ar ge -sca le b la st ing , 1.33 J/k g, w it h t he l ow er
limit strain energy for comminution from Milin s work crushing
small particles of quartz,
10
J/kg. The values differ, as could
be
expected, indicating that a large rock mass containing many joints
and pre-exis ti ng crac ks will fracture at a l ow er level of strain
e ne rg y t ha n d oe s a sma ll p art ic le of quart z. W e may c on cl ud e
t ha t t he l ow er l imi t str ai n e ne rg y
of
1.33 J/k g d er iv ed fr om t he
results of Holmberg and Persson indicates the limit where
incipient damage could be expected to occur. However, the work
of
Milin is important because it indicates a way that could
perhaps
be
used to determine the critical strain energy level from
impact experiments usin
y
larger samples of rock mas s, say in t he
range from I liter to I m 2.5 kg to 2500 kg).
o
V
E
=E= ;;
The
relationship in Equation
I
ho lds e xa ct ly for a sin e- wa ve
a nd a pp ro xi ma te ly for o th er w av e forms sim il ar to a sin e- wa ve .
It
should
be
noted, however, that the peak velocity occurs where
t he s urfa ce is at its ori ginal pos iti on this is an infl ect ion poi nt
where the surface is not bent one way
or
the other and
consequently experiences no stress
or
strain), while the velocity is
zero at the crest and at the bottom of the waves, where the
particle velocity is zero.
In
other words, the velocity and stress in
the vibration are 90 degrees out of p ha se w it h e ac h other. T hi s is
a p ro pe rt y t ha t t he sur fa ce w av e sha re s w it h all sin gl e h ar mo ni c
oscillating systems.
The strength of a rock mass is much hi gher in compression
than in tension. Th er ef ore , fract ure oc cur s in t ens ion only, in
mechanical crushing as well as in blasting.
The
local strain energy es at a given point in the rock mass is
o ne h al f of the product of t he stress a nd t he strain at that point,
thus
Fla 2 - The Rayleigh wave
In this
schematic drawing, the vertical
amplitudes have
been
exaggerated to more clearly
show
the bending of the
surface. In
reality,
the amplitude
in an
elastic wave at the elastic tensile
strength limit is
no
more than perhaps 1/1000 of the wavelength.
ratio of stress 0 to elastic modulus
E
i s a lso pr op or ti on al to t he
particle velocity and inversely proportional to the wave velocity,
so that
we
can write approximately
I
10
2
1
i
es
= 0 E = = c
2
E 2
Take as an exampl e a Rayleigh wave having a peak particle
velocity v = I rnIs a nd p ro pa ga ti ng w it h t he w av e v el oc it y CR
3 rnIs in a rock mass which has the elastic modulus =
60000
MPa. Ho lmb er g and Pe rss on 1978; 1979) ca rri ed o ut
extensive experiments in large-scale blasting, in which the
vibration particle velocities or primarily accelerations) caused by
the explosion of nearby extended charges were recorded and the
co rr es po ndi ng rock d am ag e was ma pp ed out . T he y found that
t he first mea sura bl e reducti on in s trength of the rock mass
corresponded with rock mass vibrations having particle velocities
in the range 0.7 rnIs to I rnIs The da ma ge to the rock mass
consisted of opening of p re -e xi st ing c ra ck s w hi ch r esul te d in
swelling
of
the rock mass; the swelling was measurable by
extensometers. No new cracks were formed, however, as
indicated
by
t he o bser va ti on t ha t t he re w as no d if fe re nc e in t he
RQD rock quality designation) number determined for core drill
samples of t he r oc k a t t he sam e d ista nc e f ro m t he d ri ll ho le t ak en
before and after the blast.
The
RQD-number is a measure of the
average length of u nb ro ke n d ri ll -c or es r ec ov er ed from c or e
drilling.
Fro m E qu at io n I we find the stress corresponding to the wave
v el oci ty I rnIs to be
o
= =
I
= 20 MPa
422
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A NEW TECHNIQUE FOR PREDICTING
ROCK
FRAGMENTATION
PREDICTION OF
VIBRATION VELOCITY AND
STRAIN ENERGY IN BLASTING
Holmberg and Persson 1978; 1979 found a way of calculating
the vibration particle velocity within a rock mass relatively close
to an extended charge in a blasthole resulting from the detonation
of
the charge. Figure 4 shows for two different charge
arrangements the resulting vibration velocity as a function of the
distance from the charge, with the linear charge concentration kg
explosive per m charged hole length a a parameter. The charge
arrangement shown in Figure 4a is characteristic for large hole
diameter open pit bench blasting, the charge arrangement shown
in Figure 4b is characteristic of the charges used in tunneling.
The calculated diagrams
of
vibration particle velocity versus
distance in Figure 4 have proven to be very powerful tools in
preventing damage to the remaining rock in smooth-blasting and
pre-splitting as well as in understanding the damage caused by
other techniques for perimeter blasting. The simple damage
criterion in the form
of
a critical vibration particle velocity, which
for hard igneous rock is in the range from 0.7 m/s to 1 m/s, is
used to determine whether unacceptable damage occurs or not.
The
simple criterion, although crude, has made possible a
consistent treatment
of
widely different rock damage situations.
It has been applied to control and limit damage to the remaining
rock in large open pit bench blasting as well as in tunneling and
road cuttings.
To take into consideration the effect
of
this additional free
surface, we will assume that the vibration particle velocity at a
given point in the rock to be removed is twice the calculated
velocity value at an equipositioned point in the remaining rock.
This assumption is consistent with the well-known effect
of
a free
surface which doubles the particle velocity
of
a simple
compressive elastic wave as it is reflected at the free surface as a
tensile wave. Figure 5 shows such calculated velocities behind
and in front
of
a drillhole in a bench blast ing geometry. For
simplicity in the calculations shown in Figure 5, the effect on the
vibration velocity of the free surface at the top of the bench has
been assumed not to superimpose on the effect
of
the front free
surface - possibly, the fragmentation of the rock at the corner
where the front and top free surfaces intersect may be aided by
the expansion of the rock in two directions perpendicular to each
other .
The mode of vibration
of
the rock to be removed can be
considered as the bending vibration of a prism-shaped beam of
rock, bounded by the original free surface and the cracks
extending from the drillhole at an angle towards the free surface.
Initially, the peak vibration particle velocity varies across the
thickness of the beam, but the overall effect is a translational
motion
of
the central part
of
the beam away from the charge. The
particle velocity at the ends of the beam at the intersection of
these cracks with the original free surface is less than that at the
free surface opposite the drillhole, because
of
the difference in the
FRAGMENTATION IN ROCK BLASTING
ISO-Vl:l.OClTY ONTOURS
3000 j
2.5 m/s
K - 1.4
- 0.7
1 0 - - t - - - - 1 - - - - - - - - - -
5
o
-
- - : : : : - - I e - : : : : - = ~ - -
- 5
-3 0
-3 5 -l- . . . . . . I-- -- -- ---t-- ----
-1 0 - 5 0 5 0 15 20
Olstonce
m
Fla 5 - Curves of
constant
peak
vibration
particle
velocity around
a
charge
in a drill
hole
in rock, assuming the vibration particle velocity doubles in
the rock
to be
removed
as
a result of the existence of
an
additional
free
surface the front surface of
the
bench).
Whether fragmentation will occur
or
not in rock adjacent to an
exploding charge in a drillhole in that rock depends entirely on
the presence of a pre-existing free surface. Such a surface will
provide the necessary expansion space for fragmentation, which
occurs as a result
of
tensile st resses set up by the vibration.
Since the brittle rock materials are an order of magnitude
stronger in compression than in tension, we can safely neglect
compressive stresses as a cause of fragmentation . Formally, the
method proposed by Holmberg and Persson 1978; 1979 for
calculation
of
the vibration particle velocity in the rock
surrounding the extended charge can be applied equally well on
both sides
of
the drillhole, ie to the rock which will be left
s tanding after the shot as well as to the rock which will be
removed. However, there is a major difference, in that the rock
which will be left standing will be vibrating towards only one
free surface, namely the one formed by the fractures extending
from the drillhole, whereas the rock which will be removed will
have an additional free surface allowing Vibration, namely the
original front
of
the bench. This additional free surface provides
the expansion room for the rock to be fragmented.
3000
1
1 2
Di.tonce m
b
,...
E
5
2000
:?:
u
0
u
>
c
1000
2
e
.c
>
50
0 20 30 40
Distance m
1234
6 .
~ m ~
I
1000 - - - - J - ~ - : . . . . _ - - : - - - - . . . - - i
2000
a
Fla 4 - Calculated
peak
vibration velocity
as
a
function
of distance
to a)
one end
of a
15
m
long,
large-diameter charge,
and
b) the
center ofa 3 m
long,
smaller-diameter
charge,
with the linear
charge
density as a parameter. The charge
arrangement
in
a)
is typical of bench
blasting with large diameter
holes, the arrangement in b )
is
typical for tunnel blasting and road
cuttings.
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P-A PERSSON
T LE 1
Damage andfragmentation effects in hard Scandinavian bedrock
resultingfrom vibrations with different values of the
peak
particle
velocity from Persson. Holmberg, and Lee, 1994 .
The table also includes new corresponding values of the tensile
stress.and the strain energy, calculated using Equation
1
with
E = 60000 MPa, c = 3000 m/s, and
po
= 2500 kg m
Peak particle
Tensile stress
Strain energy
Typical effect
in
velocity m/s)
MPa)
J/kg
hard
Scandinavian
bedrock
0.7
14 0.65
Incipient swelling
I
20
1.33
Incipient damage
2.5
50
8.3
Fragmentation
5 100
33
Good fragmentation
15
300 300
Crushin
SUGGESTION FO R FUTURE WORK
FIG
6a
- Th e vibrating beam bounded by the front surface
of
the bench and
the cracks extending from the drillhole at an angle to the front surface
of
the bench dashed lines indicate two sets
of
pre-exlstingjoints along
which fragmentation occurs), b) schematic picture of the resulting
fragmentation.
distance from the exploding charge in the drillhole. The resulting
bending of the beam will result in tensile stresses and
fragmentation, if the resulting stresses are large enough. The
vibrating beam is shown in Figure 6a; the resulting fragmentation
is
shown schematically in Figure 6b.
As an example, let us assume the vibration particle velocity at a
point two-thirds of the way between the drillhole and the free
surface opposite the drillhole is 15 m1s while the velocity at the
ends
of
the beam is 5
m1s.
The difference, 1
m1s
would then be
representative
of
the initial strain energy available for
fragmentation. To obtain the strain energy, we calculate the strain
as the ratio between the particle velocity 1
m1s
and the wave
velocity, which we assume to be 3000 m1s ie a strain of 1/150.
This strain energy obtained from Equation 2 with E = 60 000 MPa
is then
1
1
2 6 6 3
= 2 3000)
6 o o 1
= 0.33 * 10
m
= 133J kg
using
po =
2500 kg/m
3
for the density
of
the rock mass.
We
may
compare this value with the rock damage and fragmentation
effects at different peak vibration particle velocities tabulated by
Persson, Holmberg, and Lee 1994) as shown in Table I , where
we have included new values of the corresponding peak tensile
stress and new corresponding values
of
the strain energy,
calculated using Equation 2 with the more reasonable values
E
=
60 000 MPa, c
=
3000
m1s
and
po =
2500 kg/m
3
in their original
calculations, Persson, Holmberg, and Lee used E = 50 000 MPa
and c = 5000 m1s .
The value of the strain energy es =
133
J/kg calculated above
for the vibration velocity 1 m1s is in the region Good
fragmentation to Crushing . Comparing the strain energy es
=
133
J/kg with the data for crushing obtained by Milin 1994), we
find, not unexpectedly, that a much larger strain energy is
required for good fragmentation of small grains of quartz of the
order
of es
= 10000 J/kg) than the value
es
= 133 J/kg that we
found for the large rock mass involved in rock blasting. Again,
Milin s crushing experiments may indicate a way in which impact
crushing experiments using large samples of rock mass can be
used to determine the levels
of
strain energy that correspond to
different levels of fragmentation.
I would like to propose two types
of
experiments to further
explore the practical application potential of the strain energy
criterion for rock damage and fragmentation.
The first series of experiments would be to place
accelerometers in the rock mass to be fragmented. Even if the
time of useful recording would be limited considering the
large-scale motion of the rock in which the accelerometers are
positioned, and even if not all accelerometers could be recovered
after each experiment, the records would provide extremely
valuable information on the initial motion of the rock and
especially confirm or refute the tentative assumption that the
vibration velocity in the rock to be fragmented is twice that in the
rock that will be left standing.
The second series of experiments would involve impacting
free-standing short cylindrical lId = 1 or cubic samples of rock
by a heavy mass falling on the sample from above or suspended
in a pendulum, hitting the sample from the side. T he mass and
height of fall could be varied to vary the strain energy imparted to
the sample, and the sample size could be varied from say I litre to
I m
3
. In addition, strain gauges or accelerometers could be used
to
record the stress and strain in the ensuing vibratory motion.
Depending on the level
of
strain energy imparted, measurements
could be made
of
the swelling
of
the sample or, at higher strairr
energies, the fragment size distribution can be analysed by sieve
analysis o r by weighing fragments grouped in different size
intervals. The objective would be to establish curves similar to
those provided by Milin, but for larger samples of rock, more
closely representative
of
the size
of
the burden in rock blasting.
ACKNOWLEDGEMENTS
The author is grateful
to
Mr Mick Lownds
of
Viking Explosives,
Salt Lake City, for a short but extremely useful discussion of rock
fragmentation early in February, 1995 during a chance encounter
in the reception hall of the 16th ISEE Conference on Explosives
and Blasting. During this discussion Mr Lownds pointed out to
the author that Dr Milin s results on rock comminution
in
mechanical crushing might have r elevance to the problem of
predicting fragmentation in blasting. Subsequent reading
of
Mr
Lownds paper submitted to that conference, and discussions with
graduate student Vilem Petr
of
New Mexico Tech s Department
of Minerals and Environmental Engineering led to the thoughts
presented
in
this paper.
424
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R F R N S
Persson, P A, Holmberg, R and Lee,
J
1994.
Rock lasting and
Explosives Engineering CRC Press: Boca Raton, FL, USA) 240pp.
Holmberg, R and Persson, P A, 1978. The Swedish approach to contour
blasting, in Proceedings th Conference on Explosives and lasting
Techniques Soc
Expl Engineers: New Orleans, LA).
Holmberg, R and Persson, P A, 1979. Design tunnel perimeter
blasthole patterns to prevent rock damage, in
Proceedings Tunneling
79 Bd: M
J
Jones) Institution
Mining and Metallurgy: London).
A NEW TECHNIQUEFOR PREDICTING ROCKFRAGMENTATION
Lownds, M, 1995. Prediction
fragmentation based on distribution
explosives energy, in
Proceedings th Conference on Explosives and
lasting Techniques Int Soc Expl Engineers: Nashville, TN,
USA).
Milin, Ludovic, 1994. Incomplete Beta-function modeling
the tIo
procedure, Internal Report Public), Comminution Center,
University Utah: Salt Lake City, UT, USA).
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