1

Chapter 1. Introduction

Fragmentation produced by explosive blasting is an extremely complex process involving the nucleation and propagation of myriad microcracks that finally coalesce, breaking the rock into fragments. Moreover, the fragmentation is affected by inherent properties of the material, loading conditions, geometry such as the existence of free boundaries and discontinuities, and so on. Therefore, most blast models today depend on a suite of models and equations that are based on empirical or semi-empirical formulas. The major difficulty involved in exploring fragmentation may be that complete experimental observations are rather difficult to obtain, although some attempts have been made using high speed photography coupled with various techniques.

Fragmentation depends largely on the dynamic fracture process, which plays an important role in controlling the number of fractures produced, fracture propagation direction, etc. In order to control fragmentation, it is necessary to consider the fracture processes associated with material properties and the external force. To reveal the fragmentation mechanisms, it is first necessary to develop an appropriate method of analysis that simulates the progressive fracture of rock leading to failure and allows prediction of rock fragmentation under various loading conditions. This should enable an explanation of the mechanisms related to the dynamic strength and dynamic fracture phenomena in rock. The mechanisms are of considerable importance in understanding the fragmentation. Therefore, research on this problem is warranted. To investigate the fragmentation mechanism in bench blasting, it is also necessary to develop a new method for predicting blast fragmentation (the fragment size distribution) from the results of analysis. In addition, to understand blast fragmentation in bench blasting fully, rock fracturing due to stress waves and gas pressurization should be considered.

This dissertation proposes a dynamic fracture process analysis for simulating dynamic fracture

propagation in rock under various loading conditions. Using the proposed analysis method, the strain rate dependency mechanisms of the dynamic tensile strength of rock are revealed. The analysis is extended to clarify the dynamic fracture mechanisms involved in blast-induced borehole breakdown. To predict and control blast fragmentation, a new method using the proposed analysis method and an image analysis program is proposed. The optimum conditions for blast fragmentation with respect to blast pattern are discussed. Furthermore, to investigate the optimum conditions for blast fragmentation with respect to delay time, analysis of the dynamic fracture process is improved to simulate a fully blast-induced fracture by coupling it with a detonation gas flow model.

Summery of the Doctoral Dissertation (Sept. 2003)

Title: Dynamic Fracture Process Analysis of Rock and Its Application to Fragmentation Control in Blasting

Sang Ho Cho Supervisor: Prof. Katsuhiko Kaneko

Graduate School of Engineering, Hokkaido University, Japan

Chapter 2. Dynamic Fracture Process Analysis

Common numerical methods can be adapted

to model rock fracture in blasting, including finite element, boundary element, discrete element, and finite difference techniques. In fracture mechanics, the key point is that the propagation of cracks is affected by stress near the crack, which alters the final fracture pattern (i.e., fragmentation). Furthermore, to ensure that the numerical approximations associated with crack propagation are accurate, the mesh or grid must be refined in regions such as the crack tip. For this reason, the finite element method has been used to model crack propagation. Most methods have the capacity to compute stress intensity factors and require a complex algorithm for modeling the extension and direction of the propagation of cracks. By contrast, this study suggests a numerical model using the fracture process zone (FPZ) model that eliminated the numerical complexity that results from using a stress intensity factor. For a more reasonable simulation of the fracture process in rock, dynamic fracture process analysis incorporates rock inhomogeneity and the FPZ model. The rock fracture processes involving rock inhomogeneity

2

and the FPZ model are discussed. The proposed method is called “dynamic fracture process analysis”.

In the dynamic fracture process analysis, the increment displacement form of a dynamic finite element method is used to explain large displacement behavior. A re-meshing algorithm is used to model crack propagation, assuming that tensile fractures, i.e., crack initiation, propagation, and interconnection, occur at element boundaries. Therefore, cracks are modeled as separations from element boundaries that do not change the shape of the elements. At each element boundary, the fracture potential is checked at every time step. The fracture potential is calculated from the ratio of the normal stress and tensile strength at the element boundary. If the fracture potential of two elements exceeds 1, the node between the elements is separated into two nodes. Since the cracking and fracture processes are treated as the separation of elements, contact problems, i.e., overlapping of the separated elements, may occur due to the perpendicular compression stress that is applied to the separated elements. This problem is solved iteratively to prevent meshing overlaps when the separated elements are in contact with each other.

Finally, the Incomplete Choleski Decomposition and Conjugate Gradient (ICCG) method is used to improve the computational time. An algorithm to solve the set of linear equations is used to decrease the required computational memory.

Note that the dynamic fracture process analysis is based on the concept of fractal fracture mechanics, which supports the hierarchical process of fracture in rock. Consequently, the proposed analysis method should make it possible to explain rock fracture processes under various loading conditions. Chapter 3. Strain-Rate Dependency of

the Dynamic Tensile Strength of Rock Experimental results and discussion

An experimental approach based on Hopkinson’s effect combined with the spalling phenomena was developed to determine dynamic tensile strength. The experimental results are used to analyze and discuss the effect of the strain rate on the tensile strength in the present study. The results show a great deal of scatter in the tensile strength data at a given strain rate, especially at higher rates. The scatter of the dynamic tensile strength data in the rock specimens in particular increased significantly with the strain rate. This

discussion showed that there are different mechanisms for the strain-rate dependency of the dynamic and static tensile strengths. This study, therefore, focuses on the strain-rate dependency mechanism of the dynamic tensile strength of rock.

Fracture process simulations

The fracture processes under various loading conditions were analyzed with a proposed finite element method to verify the differences and the strain-rate dependency of the dynamic and static tensile strengths. Random numbers satisfying Weibull’s distribution were generated to give the spatial distribution of the microscopic strengths.

The fracture processes were simulated using specimens with ten different microscopic strength spatial distributions when m=5 and m=50 with the four models. The fracture processes of a specimen with m=∞, which corresponds to homogenous rock, were also simulated with each model. The maximum principal stress distribution and crack propagations in a specimen with different microscopic strength spatial distributions when m=5 are given in Figs. 3.8 (a). The compressive stress wave caused by the incident pressure reached the free end of the specimen at 56.4 µs and was reflected as a tension wave. The tensile wave was superimposed upon the tail of the compressive stress wave, and developed an increasing amount of tension. It started to produce a large number of microcracks at 61.2 µs.

Figures 3.9(a) show the final crack patterns in specimens with different microscopic strength spatial distributions using for Model I when m=5. The first fractures that were formed were selected as the fracture planes, as indicated by the triangles. It was realized that the positions of the fracture plane varied with the microscopic strength spatial distribution.

To determine the static tensile strength and the homogeneity effects, the fracture processes were simulated using specimens with ten different spatial microscopic strength distributions for both m=5 and m=50. The static tensile strength of a specimen was determined from the stress at the peak value of the stress-strain curve. Strain-rate dependency of dynamic tensile strength of rock and discussions

The strain-rate dependency of the apparent and local dynamic tensile strengths was determined from the fracture process results. Figures 3.15 show the relationship between the dynamic tensile strength and the apparent strain rate when m=5, 50 and ∞ for all the models. The apparent strain rates

3

were derived from the displacement velocities at the free end using the same approach as used in the experimental approach. The dotted lines denote the average static tensile strength in a model with the same uniformity coefficient. The dynamic tensile strength and the scatter of the strength data increased with the apparent strain rate. The apparent strain rates varied from 0.94 to 3.46 s-1. Model I showed the highest apparent strain rate, while Model IV showed the lowest apparent strain rate. The slope of the fitted line and the scatter in the strength data decreased with increasing m. Even in the homogenous models, the dynamic tensile strengths were slightly different from the

mean microscopic tensile strength, as shown in Fig. 3.15(c). Moreover, the difference increased at higher strain rates. The difference is possibly due to variations in the position of the fracture plane due to stress variations, caused by specimen surface waves that correspond to Lamb’s wave.

Different tensile fracturing processes were observed in dynamic fractures based on Hopkinson’s effect and static fractures under uni-axial tension. The differences were caused by the stress concentrations and redistribution mechanisms. However, although the mechanisms were different, the static and dynamic tensile

(a) Figure 3.9 (a) Fracture and crack patterns with

various spatial microscopic strength distributions in a Model I specimen, when m=5. The triangular symbols indicate the fractures.

Figure 3.8 (a) Distribution of the axial stress and crack propagation process in a Model I specimen

Figure 3.15 Dynamic tensile strength plotted against the apparent strain rate: (a) m=5, (b) m=50, and (c) m= ∞.

4specimen that was used for the experiments (Fig.

confirmed that the

hapter 4. Influence of the Applied W

yn

s how the bore

strengths were close to the mean microscopic strength at higher values of the uniformity coefficient as shown in Fig. 3.17. Thus, the inhomogeneity of rock contributed to the different specimen strengths under dynamic and static loading conditions. The range of the dynamic tensile strengths became narrower and the strengths were closer to the mean microscopic tensile strength as the uniformity coefficient was increased.

The fracture processes of specimens at high and low strain rates are illustrated in Figs. 3.18 (a) and (b) to verify the strain rate effect on the dynamic tensile strength. A high strain rate increased the number of microcracks, interfering with the formation of the fracture plane because of the crack arrests caused by the stress released from adjacent microcracks. Ultimately, this increased the stress without forming fractures, resulting in a high dynamic tensile strength. At lower strain rates, the number of microcracks and the number of crack arrests caused by the stresses released at adjacent microcracks were reduced by the lower strain rate, while the number of longer microcracks was increased. Ultimately, this stimulates the formation of fractures at lower applied stresses and lower dynamic tensile strengths. A fluorescent resin was used to observe the cracks and fractures in a

experimental dynamic tension tests generated numerous cracks in the test specimens, and support the numerical simulation data.

Thus, this study could verify the strain rate depe

3.19). These results

ndency mechanism of dynamic tensile strength in rock. C

Pressure aveform on the Dynamic Fracture Processes in Rock

amic Fracture Processes in RockD

The purpose of this study is to discushole pressure waveform in rock affects

Figure 3.19 Cracks and fracture plane in an Inada

t=59.5µs

t=60.5µs

t=61.5µs

t=62.5µs

t=63.5µs

granite specimen photographed with fluorescent resin emitted light.

t=58.5µs

(a)

t=141µs

t=142µs

t=143µs

t=144µs

t=145µs

t=146µs

(b)Figure 3.18 Crack nucleation, propagation,

intersections, and maximum principal distribution: (a) Model I and (b) Model IV.

Figure 3.17 Distribution of the static and dynamic tensile strengths when (a) m=5 and (b) m=50.

5

were studied using the dyna

distance was used to investigate the relationship between the applied borehole

ynamic Fracture Processes in Rock with a

The dynamic fracture process analyses were exte

dynamic fracture propagation and patterns. This includes the basic principal of dynamic fracture, which is an important factor for controlling fracture propagation and fragmentation in rock under various loading conditions.

The fracture processes (a) t0=10µs and β/α=1.5

(b) t0=100µs and β/α=1.5

(c) t0=500µs and β/α=1.5

(d) t0=1000µs and β/α=1.5

(a’) t0=10µs and β/α=100

(b’) t0=100µs and β/α=100

(c’) t0=500µs and β/α=100

(d’) t0=1000µs and β/α=100

Fig

mic fracture process analysis to investigate the influence of an applied pressure waveform on the dynamic fracture processes in rock. The fracture processes due to applied stress fields were compared for different rise and decay times as shown in Fig 4.6. Crack extensions, which corresponded to tensile fractures, increased with the rise time, and the fracture patterns were affected to a greater extent by the rise time than the decay time. The effect of the initial stress-loading rate on the fracture processes was also investigated. A higher stress-loading rate increased the number of radial cracks, thereby releasing intense levels of stress around the running cracks. The stress released from adjacent cracks affected the crack extensions and resulted in shorter crack propagation lengths. At lower stress-loading rates, the number of cracks and the crack arrests caused by the stress released at adjacent cracks were reduced. This led to longer crack extension. These analyses showed that when the preferential cracks branched earlier, the crack extension was longer.

Travel time

pressure and crack extension. It was realized that crack arrest occurred after the peak phase of stress wave in all cases. This implies that crack extension can be expected due to the applied pressure even when a longer rise time is used. This corresponds to the static fracture phenomenon in which fracture extension is dependent on the peak value of the applied pressure. Dfree face

nded to investigate the influence of the applied pressure waveform on the dynamic fracture processes in a free face model (Fig.4.13). The crack extensions increased with the rise time; if the rise time was sufficient, the crack extension was dependent predominantly on the rise time and the peak value of the applied pressure, regardless of the decay time. The fractures developed well between the borehole and the free face and parallel to the free face. Moreover, as the rise time increased fewer preferential cracks were visible towards the free face and the direction parallel to the free face.

ure 4.6 Fracture patterns with different pressure waveforms.

Figure 4.13 Fracture patterns with different pressure waveforms in the one-free-face model.

(d) t0=1000µs and β/α=1.5 (d’) t0=1000µs and β/α=100

(c) t0=500µs and β/α=1.5 (c’) t0=500µs and β/α=100

(b) t0=100µs and β/α=1.5 (b’) t0=100µs and β/α=100

(a) t0=10µs and β/α=1.5 (a’) t0=10µs and β/α=100

wave with respect to both the P-wave velocity and the burden. The transition condition from dynamic to static fractures was discussed using the non-dimensional time parameter Cp t0/w. The transition occurred between 3.0 < Cp t0/w < 4.0.

All of the analyses described above assumed that the coefficient of uniformity m was 5. Additional analyses were performed for m = 20 to examine the influence of rock inhomogeneity on fracture patterns when blasting rock with one free face. It was realized that when Cp t0/B is small, rock homogeneity can strongly affect fracture processes. The effect of rock inhomogeneity on the fracture patterns decreased as Cp t0/B increased. Chapter 5. Blast Fragmentation in

Bench Blasting Field experiments

This study focuses on estimating rock fragmentation in a field bench blasting. Sieve and image analyses were used as means of measurement. To evaluate the fragment size distribution, including fine materials, in bench blasting, two bench experiments were conducted and the blasted fragments were sieved within the range 74 µm to 1 m (Fig. 5.2). From experimental results, it was found that the size distributions obtained by image analysis did not contain the fines ratio of 13%. After the distribution corrections with the evaluation of the fines, the fragment size distributions were approximately coincident with the fragment size distributions obtained by sieving analysis and the average mean particle size became closer to the mean particle s

6

The P-wave and crack propagations shown in Figs. 4.13 (b), (c), and (d) are plotted in the same manner as in the previous model to analyze the relationship between the applied pressure and fracture extension. These are shown in Fig. 4.14. The preferential cracks, which propagated approximately parallel to the free face, were used in the analyses. The results of these analyses indicate that the reflected stress wave accelerates the speed of the running cracks and leads to greater crack extension than the circular model. Regardless of the rise time, the crack arrest time was dependent on the arrival time of the reflected

the frin the

ize obtained by sieving analysis. Gaudin-Schuhman distribution agreed with agmentation obtained by the sieving analysis range from 74 µm to 1 m approximately.

0.0 0.5 1.0 1.5 2.0 2.5 3.00

500

1000

1500

2000

2500

3000

HoleDistance from the blast hole (m)

Trav

el ti

me

(µs)

R1

R0

D1

D0

(c) t0=1000µs Crack propagation, Cc=1463 m/s

0.0 0.5 1.0 1.5 2.0 2.50

500

1500

2000

1000 R1

R0

D1

D0

(b) t0=500µs Crack propagation, Cc=1780 m/s

Hole ce fr

Trav

e (µ

s)

0.0 0.5 1.0 1.5 2.00

200

400

600

800

1000

R1

R0

D1

D0

Distan om the blast hole (m)

el ti

m

Hole Distance from

Trav

el(µ

s)

the blast hole (m)

tim

e (a) t0=100µs

Crack propagation, Cc=2329 m/s

Figure 4.14 Relationship between crack and stress wave propagation in the one-free-face model when (a) t0=100 µs, (b) t0=500 µs and (c) t0=1000 µs: R0: Front phase of reflected stress wave and R1: Peak phase of reflected stress wave

Figure 5.2 Fragment size distribution obtained inthe sieving analysis.

1E-5 1E-4 1E-3 0.01 0.1 1 100

20

40

60

80

100

Pass

ing

Perc

enta

ge (%

)

Size (m)

No. 1 No. 2

Rock fragmentation in bench blast simulation

A new numerical approach for prediction of rock fragmentation and for verifying the fragmentation mechanism in bench blasting was propo

7

rs between e holes and the free face. To investigate the

bta merical approach were ith

crease, the mean particle sizes D

to reduce the mea

es and the free face are used. Considering the bench blast simulations, using a widely spaced pattern may reduce the superposition of stress waves by increasing the travel distance from adjacent holes. To investigate a widely spaced pattern, Type 1 was

ragmentation control with respect to delay

The argument about fragmentation-related

ting that they did not consider the fracture

1E-5 1E-4 1E-3 0.01 0.1 1 100

20

40

60

80

100

Pass

ing

Perc

enta

ge (%

)

Size (m)

Type 1 Type 2 Type 3 Type 4 Type 5

sed. Five models were examined using the proposed approach in order to consider the effect of the applied borehole pressure and geometry in bench blasting (Table 5.2). From the fracture processes in the bench blast models, it was realized that the predominant fracture mechanism in simultaneous blasting were the tensile fractures (spalls) generated by the reflected tensile stresses from the superposition of radiating stress waves rom the adjacent holes, and the cratef

thinfluence of blast pattern on fragmentations in bench blast simulations, the fragmentations

ined by the nuocompared and analyzed as shown in Fig. 5.16. Wthe specific charge in

50, top sizes K and the ns values decreased, while fine ratios increase. With increasing burden and spacing distances, the mean particle sizes, top sizes K and fine ratios increased approximately, while the ns values decrease. Thus, it was realized that the only burden and spacing increase results in the uniformity decrease of the fragments. Table 5.2 Blast patterns for the bench blast simulations.

Fragmentation control with respect to the blast pattern As found in Fig. 5.16 the mean fragment size decreased with increasing the fine percentage. Here, we consider whether it is possible

n fragment size without increasing the fines percentage. In practice, a widely spaced pattern and small-diameter auxiliary holes between the hol

changed to consider a 1.5-m burden and 3.2-m spacing, to maintain the same specific charge. Figure 5.20 compares the fragmentation in the two fracture processes. The fragments with the widely spaced blast pattern were smaller than those in Type 1, except for the size in the fines distribution. These results confirm that widely spaced patterns generally produce more uniform fragmentation in bench blasting. Ftime

delay timing has not been resolved. Here, it is worth noprocess in rock, which is an inhomogeneous material, or the interaction of stress waves and crack propagation. with regard to a detonator instead of the explosive To examine the effect of delay timing, the time interval of the blastholes in Type 1 was increased to 0, 100, 500, 1000, and 2000 µs. Practically, it is very difficult to reproduce

Model Burden (m) Spacing (m) Specific charge (kg/m3)

Type 1 2.0 2.4 1.14

Type 2 2.0 2.4 1.53

Type 3 2.0 2.4 1.72

Type 4 1.5 1.8 1.53

Type 5 2.5 3 1.53

Figure 5.20 Comparison of fragmentation in general bench blast and wide space pattern.

1E-5 1E-4 1E-3 0.01 0.1 1 100

20

40

60

80

100

Pass

ing

Perc

entag

e (%

)

Size (m)

Type 1'(Wide Space) Type 1

Figure 5.16 Numerical fragmentations for all the types on a log-lin plot.

8

the sam n previously, despite using the same blast pattern at the same site, due to the inhomogeneity of rock. To consider the influence of rock mass inhomogeneity on the predicted fragmentation with various delay timings, five blast models with different microscopic distributions of strength were simulated and the mean fragment sizes are compared in Fig. 5.23. The numbers in the legend are random numbers generated to give different spatial distributions of the microscopic strength distribution. These comparisons showed that the average mean fragment size oscillated with increasing delay time, due to phase variation in the interaction of stress waves and cracks, and the superposition of stress w propo lved the argument concerning the fragmentation-related

der the action of gas pressure. The

oblem is warranted.

s

ow

e. An ex onnected cracks was devised and added. The extracted cracks are

To visualize the fracturing and gas flow due to e detonation of explosives, experiments using a

The hes ere observed at a range of

0.00 0.25 0.50 0.75 1.00 1.25 1.500.15

0.20

0.25

e fragmentation see

aves with increasing delay time. Therefore, thesed numerical approach reso

delay time. The damage model used in previous studies of fragmentation is not appropriate for estimating fragmentation in bench blasting.

The optimum time interval is of millisecond order. From the viewpoint of controlled fragmentation with respect to the optimum time interval, this implies that a different fragmentation mechanism has to be considered. It is well known that stress waves generate blast-induced fractures,

hich extend un

0.30

wbench blast simulations in this study did not consider gas pressurization. Therefore, research on his prt

Chapter 6. Dynamic Fracture Process

due to Stress Wave and Gas Pressurization

Numerical analysis method for modeling gas flow

This study proposes a numerical method to

saved in the network model, which is placed between the dynamic fracture process and finite difference analyses.

To consider the gas flowing through the fractures, one-dimensional transient flow, governed by the conservation of mass and momentum, was used. The flow was calculated using finite difference calculation, assuming turbulent and laminar flow.

Mea

n fra

gmen

t size

(D50

)

Delay Time/Burden (ms/m)

simulate fracture propagation due to stress waveand gas pressurization to investigate fracturing due to the detonation of explosives and gas flowing through the fractures. The method couples the finite element and finite difference methods. Anoutline of the flow diagram used to couple gas flwith the dynamic fracture process analysis is shown in Fig. 6.1. The dynamic fracture process analysis provides information about the fracture geometry: COD, crack length, and initial borehole pressure. Since the detonation gases in the borehole can flow only through the cracks connected to thehole, unconnected fractures are neglected in the gasflow calculation. Therefore, it is necessary to check whether each crack is connected to the borehol

traction algorithm for the c

Experimental observation of detonation gas flow

thhigh-speed digital video camera system, which was utilized on one side of the notches, were performed.

average gas-propagation velocity was 178 m/s. e gas velocities wT

Case 1 Case 2 Case 3 Case 4 Case 5

Figure 5.23 Variation of average size with the ratio of delay time and burden.

Dynamic Finite Element Analysis

Network Model for the Fractures (Crack search algorithm)

Finite Difference Analysis for Gas flow

Gas pressure

Figure 6.1 Flow chart of gas flow coupled with the finite element analysis.

Fracture geometry (COD, etc.)

9

of

Fracture proce wave and gas pressurization

The numerical gas-propagation velocity along the fractures dramatically dropped at near the blast hole and remained stable at about 150 m/s within a radiu average gas-propagation velocit ately agrees rical velocity at the observ 0.05m in radius. It was confirmed that the detonation gas considerably af he fractures. The gas pressures in the fractures

radually increased with time and the maximu

urization on the fracture process, the fracture roc

h the observed gas velocity. These results provip pressu

a

exper firmed previously, and showed that, in any case, once the

n of those same two

ow and stress wave from e a

about 0.03 ~ 0.09 m in radius. These measurements showed that the average-propagation velocity the stress wave was 4450 m/s.

ss analyses due to stress

s of 0.06-0.1 m. The experimentaly, 178 m/s, approxim

with the numeion point at

fects the generation and propagation of t

g mpressures were constant near the blast hole.

To investigate the influence of gas ressp

p ess resulting from borehole pressure without gas flowing through the fractures was investigated. Gas pressurization caused the fractures to extend, and the stress field differed significantly. Figure 6.25 compares the crack opening displacement

(COD) resulting from the fracture process. The gas pressurization affected the COD.

P-wave causes two radial cracks at approximately 20 to 30 degrees to the bench face to advance ahead of the others, the gas pressure loading acts to greatly favor further propagatio

Practically, fractures arising from thedetonation of explosive have various roughnesses;the gas flow also varies with the roughness. Toinvestigate the effect of fracture roughness on the gas flow, the relative roughness, ε/h, was variedfrom 1 to 3. The gas propagation velocities were compared with the experimental result, as shown in Fig. 6.27. The numerical gas velocity when ε/h =1 agreed well wit

de insight into the complete fracture rocess that is due to both the stress wave and gas

rization.

Optimum fragmentation with respect to delaytime

Chapter 5 discussed the effect of delay timingat adjacent holes. Practically, the optimum timeinterval is of millisecond order. The previouschapter showed that boulders are produced between adjacent holes and the free face. In bench blasting, the existence of the free face plays animportant role in causing the predominant fractures to be the crater and reflected wave induced-tensilefractures. Cracks traveling at an angle of 40 – 80degrees to the normal of the free surface have a greater propagation velocity. The angle increaseswith increasing burden. These fracture phenomen

imentally and numerically were con

cracks. Figure 6.28 shows the interaction of crack

extension, due to the gas flth djacent hole (Hole 1), with shot delay time. Assuming that the crater is completely separated by the gas pressurization, the crack forms a new free face. It is conceivable that after a time delay, the crack that is due to gas pressurization plays a role at a free face, and interacts with the stress wave radiating from the adjacent hole (Hole 2 in Fig. 6.28). It is possible that effective fragmentation caused by a free face effect may be expected.

From the relation between gas flow and radiating stress wave from adjacent holes, the optimum fragmentation condition can be represented as the relationship between the ratios Dt/B and S/B. Figure 6.29 shows the variation in the optimum fragmentation condition with various gas flow velocities, Cg. As an example, let us refer to Type 1 in the previous chapter and consider Cg =

0 50 100 150 200 250 3000.00

0.01

0.02

0.03

0.04

0.05

Dista

nce f

rom

the t

ip o

f not

ch (m

)

Time (µs)

ε/h = 1 (Cg= 175 m/s)

ε/h = 2 (Cg= 160 m/s)

ε/h = 3 (Cg= 151 m/s) Experiment (D.I.=6.06)

Figure 6.27 Comparison of the calculated and experimental gas propagation. velocities

(b)

(a)

Figure 6.25 Comparison of the crack opening displacement of fractures due to (a) a stress wave only and (b) both a stress wave and gas pressurization at 900 µs.

10

e d n mechacontributed to the dif en the dynamic and static tensile strengths. An increase in the

C ocess analyses for different waveforms of borehole

rack extension. These analyses revealed that the

ential crack branching occurred, the . The dynamic

fract

178 m/s, which corresponds to the experimental result in this chapter. The optimum delay time for Type I was 3.1 ms/m, which is within the range of optimum delay times determined in the field bench blast tests. Ultimately, optimal fragmentation in the field with respect to delay time depends strongly on the gas flow through the fractures caused by the stress wave.

This gives a significant insight to predict precise fragmentation control with respect to delay time

Figure 6.29 Variation in the optimum fragmentation condition with various gas flow velocities.

in field blast.

Chapter 7. Conclusions

This dissertation covered dynamic fracture

processes from basic principals to engineering applications. The findings of this dissertation should contribute to a better understanding of the rock fracture mechanics associated with dynamic fracture; they should also play a decisive role in the precise control of fragmentation in rock blasting. The contents and findings of this dissertation are summarized as follows:

Chapter 2 proposed dynamic fracture process analysis, based on the concept of rock fractal fracture mechanics, to simulate dynamic fracturing

in rock. The analysis incorporated the dynamic finite element method and non-linear fracture mechanics.

pressure in order to verify the dynamic fracture mechanism related to blast-induced borehole breakdown. The fracture processes were affected more by the rise time increases than by the decay time. A higher stress-loading rate increased the number of radial cracks and led to intense stress release around running cracks. The stress release caused by adjacent cracks interfered with crack exte

Chapter 3 examined the dynamic and static tensile strengths of rocks, and the differencesbetween static and dynamic tensile strength; the strain-rate dependency of the dynamic tensile strength was also investigated. Fracture processes under various loading conditions were analyzed using a proposed finite element method to verify the differences between the dynamic and static tensile strengths and the strain-rate dependency. These analyses revealed that the differences wer

ue to stress concentration and redistributionisms in rock. Rock inhomogeneity also

ference betwe

uniformity coefficient led to a reduction in the strain-rate dependency; i.e., the strain-rate dependency of the dynamic tensile strength was caused by the inhomogeneity of the rock. The fracture processes and principal stress fields in specimens under high and low strain rates were analyzed to investigate fracture formation at various strain rates. Higher strain rates generated a large number of microcracks; the interactions between the microcracks interfered with the formation of a fracture plane. The observed dynamic tensile strength increase at a high strain rate was caused by crack arrest due to the generation of a large number of microcracks.

hapter 4 conducted dynamic fracture pr

nsion and resulted in shorter crack propagation. At lower stress-loading rates, the number of cracks and the crack arrest caused by the stress released at adjacent cracks were reduced, leading to longerc earlier prefergreater the extension of the crack

ure process analyses were extended to investigate the influence of the waveform of applied pressure on the dynamic fracture process in a free face model. These fracture processes revealed that crack extension increased with the rise time increase, and that when the rise time is sufficiently long, crack extension depends, predominantly, on the rise time and the peak value of applied pressure, regardless of the decay time.

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

1.0

2.0

3.0

4.0

7.0

5.0

6.0

8.0

9.0

10.0 Cg = 100 m/s Cg = 150 m/s C = 20g 0 m/s Cg = 300 m/s

Dt/B

(ms/m

)

S/B ratio

Figure 6.28 Scheme diagram representing the interaction between gas flow and the stress wave from the adjacent hole.

11

ter the peak phase of the stres

; the fragment sizes of blasted rocks were

nsile fract

appr

It was realized that the optimal field fragm

3)

erground, Vol. 13,

Crack arrest occurred afs wave in all cases. The transition condition

from dynamic to static fractures was discussed using the non-dimensional time parameter Cp t0/B. The transition occurred between 3.0 < Cp t0/B < 4.0.

Chapter 5 contained the results of two bench experiments

estimated using sieving analysis and image analysis to evaluate rock fragmentation in a field bench blast. The fragment size distributions produced in image analyses were corrected for fines.

The experimental results showed that the size distributions obtained using image analysis did not contain a fines ratio of 13%. After correcting the distribution for the fines, the fragment size distribution approximated to the fragment size distribution obtained using sieving analysis, and the average mean particle size approached the mean particle size obtained in the sieving analysis. Gaudin-Schuhman distribution agreed approximately with the fragmentation obtained in the sieving analysis for the range 74 µm to 1 m. The findings show that rock fragmentation cannot be predicted simply by theory, or by an equation, because the fragmentation in bench blasting involves a wide range of size from fine to coarse.

A new numerical approach for predicting rock fragmentation and for verifying the fracture mechanism in bench blasting was proposed. Five models were used to consider the effect of the applied borehole pressure and geometry in bench blasting. The fracture processes in the bench blast models demonstrated that the predominant fracture mechanism in simultaneous blasting was the te

ures generated by the superposition of reflected tensile stresses and radiating stress waves from adjacent holes, and the craters between the holes and the free face. To investigate the influence of the blast pattern on fragmentation in bench blast simulations, the fragmentation obtained using the numerical approach was compared and analyzed. Increasing the specific charge decreased the mean particle size D50, top size K, and ns, while the fines ratio increased. Increasing the burden and spacing, increased the mean particle size D50, top size K, and fines ratio, while ns decreased. Consequently, only an increase in the burden and spacing decreases fragment uniformity.

Chapter 6 investigated the effect of gas flow though the fractures, with the detonation of an explosive, on the fracture process of rock. In order to simulate gas flow and pressurization in fractures, the dynamic fracture process analysis was combined with a gas flow model, which was based on the finite difference method. It was confirmed that the detonation gas had a considerable effect on

the generation and propagation of fractures. The gas pressure in the fractures gradually increased with time and the maximum pressures were constant near the blast hole. The numerical gas-propagation velocity along the fractures dropped dramatically near the blast hole and remained stable at about 150 m/s within a radius of 0.06-0.1 m. The experimental average gas-propagation velocity, 178 m/s, agreed

oximately with the numerical velocity at observation points within a radius of 0.05 m.

The optimum fragmentation condition associated with the delay time in bench blasting was discussed using the relationship between gas flow and the radiating stress waves from adjacent holes.

entation depends strongly on the gas flow velocity through fractures that are due to the stress wave. Publications related to this dissertation:

1) Kaneko Katsuhiko and Cho Sang Ho, 2002, Fragmentation process of rock in blasting, Japanese Journal of Multiphase Flow, Vol. 16 No.4, 345-352, (in Japanese)

2) Cho Sang Ho, Nishi Masaaki, Yamamoto Masaaki and Kaneko Katsuhiko, 2003, Fragment size distribution in blasting, Material Transactions, Vol. 44, No. 5 pp. 1-6

Cho Sang Ho, Nohara Sayaka and Kaneko Katsuhiko, 2003, Numerical approach for strain rate dependency of the dynamic tensile strength of rock. Science and Technology of Energetic Material, Vol. 64, No. 2, pp. 87-96

4) Cho Sang Ho, Ogata Yuji and Kaneko Katsuhiko, 2003, Strain rate dependency of the dynamic tensile strength of rock. International Journal of Rock Mechanics and Mining Science, Vol. 40, No. 5, pp. 763-777

5) Cho Sang Ho, Miyake Hidekatsu, Kimura Tetsu and Kaneko Katsuhiko, 2003, Effect of the waveforms of applied pressure on rock fracture mechanics in one free face, Science and Technology of Energetic Material, Vol. 64, No. 3, pp. 116-125

6) Cho Sang Ho, Yang Hyung Sik and Kaneko Katsuhiko, 2003, Influence of inhomogeneity on the static tensile strength of rock, Tunnel and UndNo. 2, pp. 1-5

7) Cho Sang Ho, Yang Hyung Sik and Kaneko Katsuhiko, 2003, Influence of inhomogeneity on the dynamic tensile

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strength of rock, Tunnel and Underground, Vol. 13, No. 3, pp. 180-186

8) Cho Sang Ho, Nakamura Yuichi and Kaneko Katsuhiko, 2004, Dynamic fracture process of rock subjected to stress wave and gas pressurization, International Journal of Rock Mechanics and Mining Science (in press)

9) Cho Sang Ho and Kaneko Katsuhiko, 2004, Rock Fragmentation Control in Blasting, Material Transactions (in press)

10) Cho Sang Ho and Kaneko Katsuhiko, 2003, Influence of the applied pressure waveform on the dynamic fracture processes in rock, International Journal of Rock Mechand Mining Science (currently submitted)