a new technique for predicting rock fragmentation in blasting

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.

EXPlO

95 Conference

Brisbane, 4 - 7 September

995 4


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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

Brisbane 4 - 7 September 1995 XPlO 95 Conference


3/6

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.

EXPLO 95 Conference

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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

Brisbane 4 - 7 September 1995

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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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a new technique for predicting rock fragmentation in blasting

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