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International Journal of Rock Mechanics & Mining Sciences 44 (2007) 412–424

Numerical investigation of blasting-induced crack initiation andpropagation in rocks

Zheming Zhua,�, Bibhu Mohantyb, Heping Xiea

aCollege of Architecture and Environment, Sichuan University, Chengdu, Sichuan 610065, ChinabLassonde Institute, Department of Civil Engineering, University of Toronto, Toronto, Ont., Canada M5S 1A4

Received 4 May 2006; received in revised form 16 July 2006; accepted 12 September 2006

Available online 31 October 2006

Abstract

To investigate the dynamic fracture mechanism related to blast-induced borehole breakdown and crack propagation, circular rock

models containing a single centrally located source of explosive were numerically blasted using the AUTODYN 2D code. According to

the material properties and loading conditions, four kinds of equations of state, linear, shock, compaction and ideal gas, are used.

A modified principal stress failure criterion is applied to determining material status. The dynamic stresses at the selected target points in

a rock sample are computed as a function of time following application of explosive load. It is shown that shear stress (resulting from

intense compressive stress) causes a crushed zone near the borehole, the major tensile principal stress causes radial cracks, and the

reflected stress wave from free boundary causes circumferential cracks some distance away from the free boundary. The influences of the

factors of boundary condition, coupling medium, borehole diameter, decoupling and joint on rock dynamic fracture are discussed.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Dynamic fracture; Blasting; Numerical model; Stress wave; Crack

1. Introduction

Fragmentation of rock by explosives remains central toall large-scale mining, quarrying and civil constructionexcavations. With increasing scale of these operations,proper design of blasts and control as well as prediction ofblast results has become imperative in most operations.However, our understanding of the blasting process is farfrom complete, as both the commercial explosives and thetarget rock are complex materials. The energy releasecharacteristics in the former are highly variable, dependingon the prevailing field parameters such as boreholediameter, density gradient, and sympathetic pressuresamong explosive columns from detonation of a neighbour-ing hole. Similarly, the response of the target rock to highdynamic loadings, which may last only for a fewmilliseconds, remains largely unknown. Under this scenar-io, the best approach to the study is to first generate an

extensive experimental database on these properties. At thesame time, it is essential to investigate the fracture andfragmentation mechanism through numerical models so asto obtain a better understanding of the dominantparameters that control blast results. The present paperdeals with this latter topic.In rock blasting, it is generally agreed that two types of

loading operate on the surrounding rock: stress wave (orshock wave) loading and explosion gas pressure loading. Thestress wave loading arises out of detonation of the explosivecolumn in the borehole. For typical commercial explosives,the detonation pressure exerted on the borehole wall at themoment of initiation could easily exceed 10GPa. This highpressure on the borehole wall sets off a shock wave in theadjacent rock mass, but it soon decays to a high amplitudestress wave propagating at the velocity of longitudinal wavein the rock mass. It is immediately followed, albeit at a muchreduced velocity, by the longer duration gas pressure loading.This loading due to gas expansion continues long after thestress wave has dissipated, as its expansion rate is con-siderably lower than that of the propagating stress waves

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1365-1609/$ - see front matter r 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijrmms.2006.09.002

�Corresponding author. Tel./fax: +86 416 978 5969.

E-mail address: zhemingzhu@hotmail.com (Z. Zhu).

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[1–3]. The former initiates cracks around the borehole, andthe latter penetrates into these cracks and causes their furtherextension and propagation. Therefore, both loadings play avery important role in efficient use of the explosive energy.

Theoretical studies on rock blasting [3–5] generallyagrees with the crack extension data; however, the mainfocus has been on the propagation of the pre-existingcracks under the gas pressure loading and lesser attentionhas been paid to the initial cracks resulting from stresswave loading. Numerical methods have been employed byseveral researchers [6–11], using various numerical codes tosimulate fragmentation process in blasting. Although manysignificant results have been published, it is far fromcomplete for the numerical study of rock fragmentation.

The objective of this numerical study is to analyze rockfracture mechanisms and explain how the crushed zone,severely fractured zone, and incipiently cracked zone arecreated under stress wave loading. This study is veryimportant because fracturing under stress wave loading isconsidered the first phase in the fragmentation process, andis the crucial stage as the subsequent fragmentation andlarge-scale movement of the fractured rock mass due tocontinuing penetration of the explosion gases are largelyguided by this initial fractured state. Therefore, this studyis a critical step in our understanding of rock fragmenta-tion by blasting.

In this numerical study, the following topics arediscussed: (a) material statuses of rock sample as a functionof time from the exploding source, (b) rock fracturingmechanism under dynamic loading, and (c) factors thatinfluence crack initiation and propagation. In order tominimize the various variables and the associated uncer-tainties of the explosive and rock, a well-behaved explosive,PETN, whose detonation properties are well known, and arelatively homogeneous igneous rock, diorite, for whichstandard strength properties have been determined, areemployed in this study. Throughout this work, tensilestress is positive and compressive stress is negative.

2. Calculation model and numerical code

In this paper, a 2D plane symmetrical dynamicnumerical model with four kinds of equations of state(EOS) and a modified principal stress failure criterion areapplied. The numerical code used in this simulation isAUTODYN code.

2.1. Calculation model and material parameters

The target rock is modeled by a circular rock samplecontaining a single centrally located source of explosive.The source of explosive is represented by a detonating cordcontaining a core load of PETN explosive (1.1 g/m and1.08mm in diameter) surrounded by a thin sheath ofpolyethylene, with the total diameter of the cord being2.36mm. The borehole diameter is 10mm, and the rocksample measures from 100 to 400mm in diameter. The

coupling materials of water, sand and air are used in thissimulation. Since the objective of the investigation is tostudy shock or stress wave induced cracking in rock,coupling materials filled in the place between the explosivecharge and the borehole wall are necessary, since it couldgreatly reduces the effect of the subsequent high-pressureexplosion gases on further development of the cracksinduced by stress waves. Any further increase in theexplosive charge diameter would have resulted in rockserious fragmentation, which is not convenient for thestudy of blasting-induced cracking.The model shown in Fig. 1 is a 2D plane symmetrical

(plane strain) dynamic model. A diorite rock was selectedas the target rock with the following properties [12]:density ¼ 3:16 g=cc, P-wave velocity ¼ 5:77 km=s, tensilestrength ¼ 18:8MPa, bulk modulus ¼ 52:4GPa, shearmodulus ¼ 39:6GPa, shear strength ¼ 44:16MPa. How-ever, it is known that dynamic fracture simulation underhigh strain rate requires ‘dynamic’ strength value, and thisvalue should increase with increasing strain rate. Butcurrently it is very difficult to use a variable dynamicstrength in AUTODYN code and also the relationshipbetween dynamic strength and strain rate for this rock isnot available. Thus, we have to select a constant dynamicstrength (since they are not presently available). It is knownthat the dynamic strength could exceed its static counter-part by nearly an order of magnitude [13,14]. But if thedynamic strength is selected too low, the correspondingrock sample will be fragmented seriously, and if it is toohigh, the damage extent will be very slight. Both of thesetwo cases are not convenient for the study of blasting-induced cracking. Thus, in this simulation the dynamicstrength is taken to be six times their static values, i.e., thedynamic tensile strength is 112.8MPa and the dynamicshear strength is 265MPa. In the following simulations, theparameters are the same unless specified otherwise.

2.2. Equation of state (EOS)

In this numerical study, four kinds of EOS, linear, shock,compaction and ideal gas, are applied. Because of thecoupling medium applied, the resulting pressure on thetarget rock is relatively low, the calculated pressure on the

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Fig. 1. Schematic of a circular rock sample containing a centrally located

detonating cord and coupling medium; where D is rock sample diameter

(100–400mm), the unit is mm.

Z. Zhu et al. / International Journal of Rock Mechanics & Mining Sciences 44 (2007) 412–424 413

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borehole wall from detonation of the explosive cord is lessthan 1.6GPa (usually, under shock wave loading thepressure is about 10GPa), therefore, for the target rock,linear EOS, which is suitable for small deformation (orsmall compression) dynamic problems [15], is employed

p ¼ k½ðr=r0Þ � 1� ¼ km ¼ 52:4m, (1)

where p is the pressure, k is the bulk modulus,m ¼ ðr=r0Þ � 1, and r=r0 is the ratio of the density ofcurrent state to that in the initial state.

For the materials of polyethylene (sheath of thedetonating cord) and water (coupling material) in theimmediate vicinity of the explosive charge, a shock EOS isemployed:

U s ¼ C0 þ S1up þ S2u2p, (2)

where C0, S1 and S2 are constants, Us is the shock wavevelocity, and up is the particle velocity. For polyethylenematerial, C0 ¼ 2901m=s, S1 ¼ 1:481, S2 ¼ 0:0, and forwater material, C0 ¼ 1647m=s, S1 ¼ 1:921, S2 ¼ 0:0 [15].

For the coupling material of sand, a compaction EOS isapplied which is more suitable for porous materials. Therelation between pressure and material density is obtainedfrom the experimental results of [16].

For the coupling material of air, ideal gas EOS isapplied, which is

p ¼ 0:4re, (3)

where p is pressure, r is density, and e is the internalenergy, e ¼ 717:3T , where T is temperature.

2.3. Failure criterion

During the process of stress wave propagation, tensilestresses or shear stresses do occur and cause rock materialto fail in tension or in shear, therefore, a modified principalstress failure criterion is applied to determining materialstatus, which is suitable for describing material tensilefailure or shear failure. The modified principal stress failurecriterion dictates that when the major principal stress orthe maximum shear stress in an element exceeds materialtensile or shear strength, the element fails. After an elementhas failed, it will not be able to sustain any tensile andshear loadings, but it is still able to sustain compressiveloading [15]. The normal compressive stresses, sx and sy, ofa failed element must be identical, i.e., sx ¼ sy. This isbecause the failed element cannot sustain any shear stressðt ¼ 0Þ, the corresponding Mohr circle will become a point.

2.4. Sub-grid and meshing

For each material, one sub-grid is defined, so in totalthere are four sub-grids in this numerical simulation (fourmaterials). Each sub-grid is covered by a set of quad-rilateral elements (or cells). The index space of each sub-grid is independent of any other sub-grids defined in thisproblem. The positions, velocities and accelerations are

defined at the nodes of elements. Pressure, internal energy,density, stress, strain rate, temperature and the mass of theelement are defined at the element centers. The nodes movewith their velocities, and the elements deform according tothe movement of the nodes.In the model shown in Fig. 1, the total number of

elements for the rock material with 100mm in diameter is32 000 and the mesh is shown in Fig. 2. The meshes for thecoupling material and the polyethylene material aredesigned similar to the mesh of the rock material, andtheir total numbers of elements are 5184 and 2040,respectively. The total number of elements for the explosivematerial is 1296. These meshes are seen to be fine enoughfor the required precision of calculation.

2.5. Numerical code

AUTODYN code is applied in this study, which can beapplied to solve a wide variety of non-linear problems insolid, fluid and gas dynamics. AUTODYN code has beensuccessfully applied in the study of rock fracture by Chenand Zhao [17] and Ma et al. [9]. The series of calculationsthat are carried out in each incremental timestep (or cycle)in a Lagrange sub-grid are shown schematically in Fig. 3.Starting at the bottom, the boundary and/or interactiveforces are updated and combined with the forces for innerzones computed during the previous time cycle. Then forall non-interactive Lagrangian nodes the accelerations,

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Fig. 2. Mesh of the rock sample (outer diameter ¼ 100mm) shown in Fig.

1. Element number is 32 000.

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velocities and displacements are computed from themomentum equation and its further integration. Fromthese values the new zonal volumes and strain rates can becalculated. With the use of a material model together withthe energy equation, the zonal pressure, stress and energycan be calculated, providing forces for use at the start ofthe next integration cycle.

In this study, the Lagrange processor is applied, and onlyfor the case of air coupling, Euler processor is applied,which is ideally suited to simulating the problems of fluidand gas flow. Additional details on the interactionsbetween sub-grids, such as Lagrange–Lagrange and La-grange–Euler coupling can be found in Refs. [15,18]. Thegoverning equations applied in AUTODYN code are massconservation, momentum conservation, and energy con-servation [15].

3. Simulation results

Based on the suggested model, EOS, failure criterion andrelated parameters, the calculations are implemented byapplying the AUTODYN code in order to investigate someof the key aspects of the dynamic fracture process in rockduring the first critical stages of blasting action, beforelarge-scale penetration of the explosion gases into thesefractures, resulting in further fragmentation and rock massmovement.

3.1. Material status as a function of time after initiation of

explosive

In order to illustrate the mechanism of cracking of rockmaterial under dynamic loading and the effect of freesurfaces, the 2D plane symmetrical model shown in Fig. 1is applied. Fig. 4 shows the material statuses of a rocksample ðdiameter ¼ 100mmÞ as a function of time after

initiation of the explosive in the borehole. Six target pointsare selected in order to analyze rock fracture mechanismlater. Due to the high radial compressive stresses near theborehole, an intense shear stress field develops near theborehole. In the shear failure zone, the rock is severelycrushed and the width of this crushed zone is approxi-mately equal to the borehole radius. The crushed zone isfollowed by a severely fractured zone. In this zone thecrack density is very high and the damage is severe. Beyondthe severely fractured zone is the incipiently cracked zone.In this zone, the radial crack density is lower than that inthe severely fractured zone. Additionally, the action of thereflected stress waves from the free surface causescircumferential cracks, which are more commonly knownas spalling cracks. It should be noted that the extent ofspalling would depend on the dynamic tensile strength ofthe target rock and the distance traveled by the stress wave.For a very large diameter sample or with a much highertensile strength, it is possible not to have any spallingfractures (that is why the rock sample diameter selected inthis calculation is small).The stress waves (or the reflected stress waves) weaken in

amplitude with time, and their role on further fragmenta-tion process is progressively replaced by the expansion ofthe explosion gases into these cracks. This is evidenced bythe continually expanding borehole, whose diameter at thetime of 100 ms (microsecond) is 31% larger than its originaldiameter (Fig. 4).In Fig. 4, the radial cracks thicken near the outer

boundary, which is caused by the element size increasing.This does not mean that the crack opening displacement isexpending.

3.2. Shear stress causing crushed failure

Target 1 shown in Fig. 4 ðradius ¼ 5:15mmÞ is near theborehole and in the shear crushed zone. The radial stress srand tangential stress sy of target 1 shown in Fig. 5 arealways compressive. After the stress wave starts to impactthe borehole, the element containing target 1 is subjected toa rapidly increasing compressive stress sr and a somewhatlower compressive stress sy. The maximum differencebetween sr and sy is 441.2MPa ( ¼ 577.5–136.3MPa) atthe time of 2.00225 ms. But just in next time step(2.00314 ms), the difference disappears and they areidentical. This is because sr is the stress resulting fromthe stress wave directly impacting the borehole wall, but syis the tangential stress and it is passive. Therefore, at thebeginning the increasing speed of sy is lower than that ofsr, which causes the difference between the normal stresses(sr and sy), resulting in shear stress (if sr ¼ sy, shear stressmust be zero according to Mohr circle). The maximumshear stress t12ðt12 ¼ ðs1 � s3Þ=2Þ is 316.0MPa, and since itis larger than the dynamic shear strength, 265MPa, theelement containing target 1 fails in shear. After the elementhas failed, the shear stresses (t12 and try) fall to zero at2.00314 ms (one time step from 2.00225 ms), and the normal

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Nodal forces

Nodal acceleration

Nodal velocity& displacement

Zone volume& strain rate

Zone pressure& Stress

Boundary and / or interactive forces

Force / mass

Integration

Direct calculation

Material model

Conservation of momentum

Fig. 3. Lagrange computation cycle in AUTODYN code.

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stresses, sr and sy, are identical in accordance with thefailure criterion applied in this numerical model. Afterfailure, the normal stresses have experienced a very highcompressive stress, 1.3269GPa. The corresponding curveof strain rate versus time is presented in Fig. 6.

3.3. Major tensile stress causing radial cracks

Target 2 shown in Fig. 4 ðradius ¼ 12:114mmÞ is in theseverely fractured zone and a crack is seen to pass throughit. The radial stress sr at target 2 shown in Fig. 7 is alwayscompressive, therefore, no circumferential cracks occur.The tangential stress sy is first compressive and then itchanges to tensile and reaches the maximum value of111.37MPa at the time of 4.2208 ms. Meanwhile the majorprincipal stress s1 is 113.36MPa ðsy � s1Þ, which is largerthan the rock tensile strength, 112.8MPa, therefore, the

element containing target 2 fails, resulting in the radialcrack passing through it. After the element has failed, dueto the application of the modified principal stress failurecriterion, the shear stress is zero, and the normal stresses srand sy are identical.The time from the stress wave fully acting on target 2

(i.e., when sr reaches its maximum value) to sy reaching itsmaximum tensile stress is 0.8853 ms (Fig. 7), that is, theradial crack occurs at 0.8853 ms delayed after the stresswave had arrived. During this time period, the stress wavefront will propagate to a distance, 0:8853ms� 5:77mm=ms¼ 5:108mm. Thus, the radial crack at target 2 is initiatedafter the stress wave front has traveled to a distance of5.108mm (see Fig. 8). This phenomenon may be physicallyexplained as follows: when the stress wave just reachestarget 2, due to the resistance from its adjacent elements,the tangential stress is compressive at the beginning. But

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Fig. 4. Material status of a rock sample ðdiameter ¼ 100mmÞ as a function of time after initiation of the explosive in the borehole; six target points are

selected in order to analyze rock fracturing mechanism.

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after the stress wave front has traveled to a certaindistance, the resistance from its adjacent elements mayrelease, causing the tangential stress sy to change from

compressive to tensile, resulting in the formation of theradial cracks. Therefore, the radial crack may occur onlyafter the stress wave has traveled to a certain distance. Thisindicates that stress wave propagation velocity is muchfaster than the speed of crack formation, which is3.203 km/s from this calculation. It should be noted thatthe cracks created by the stress wave loading are stillclosed. After the explosion gas penetrates into these cracks,they will start to open and free surface will occur.Therefore, the speed of crack formation here should notbe related to the Rayleigh wave velocity.

3.4. Reflected stress wave causing circumferential cracks

When the stress wave reaches a free boundary, the stresswave will reflect back, causing circumferential cracks. Fig.9 shows the relationship of stresses versus time for theelement containing target 4 ðradius ¼ 40:035mmÞ. Theradial stress sr before 11.662 ms is compressive, and after11.662 ms it is tensile because the reflected stress wavesuperposes with the radial stress wave. According to the P-wave velocity, the stress wave reflects back to target 4 needs11.517 ms after initiation. At the time of 12.102 ms, srreaches its maximum value of 112.58MPa, and thecorresponding major principal stress s1 is 112.85MPaðsr � s1Þ which is higher than the rock dynamic tensilestrength, 112.8MPa, therefore, the element containingtarget 4 fails and a circumferential crack develops. Mean-while, the tangential stress sy at the time of 12.102 ms is alsotensile and it is 98.531MPa. This indicates if this elementwas not first failed due to sr, it could fail resulting from thetangential stress sy.

3.5. Stress wave at free boundary

Target 5 shown in Fig. 4 ðradius ¼ 50mmÞ is at the outerboundary of the rock sample and the relationship ofstresses versus time of target 5 is shown in Fig. 10. After thestress wave reaches the free boundary, the radial stress srdue to superposition with the reflected stress wave changesbetween compression and tension for several times. Thetangential stress sy at the time of 13.502 ms, reaches itsmaximum value of 113.04MPa, which is higher than therock tensile strength, 112.8MPa, therefore, the elementcontaining target 5 fails in tension, and the radial crackreaches the free boundary.

3.6. Discussion

From Fig. 4, it can be seen that the cracks are not exactlysymmetrical. This is because the interaction logic appliedbetween two contact materials and the failure criterionapplied in this study.Fig. 11 shows the interaction gap between two contact

materials. The distance of nodes 1–3 to the boundary ofmaterial 2 could be slightly different. This will result indifferent nodal forces repelled as they enter the contact

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Fig. 5. Relationship of dynamic stresses (i.e., radial stress sr, tangentialstress sy, shear stress try and maximum shear stress t12) versus time for the

element containing target 1 ðradius ¼ 5:15mmÞ.

Fig. 6. Relationship of strain rate versus time for the element containing

target 1.

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detection zone according to the interaction logic applied inAUTODYN code [15], therefore, the element stresses couldbe slightly different. The finer mesh, the less the differencewill be. But when the number of elements exceeds a certainvalue, the influence of the number on the calculation resultsdiminishes. In this simulation, along the borehole (5mm inradius) there are 320 elements, which is fine enough. When

the element number increases from 320 to 480, thedifference of the calculation results is small.According to the failure criterion applied, if an element is

failed, it will not be able to sustain any tensile and shearloadings, i.e., its tensile and shear stresses will release,which will significantly affect the stress field around thisfailed element. Fig. 4 shows that target 3 is failed, buttarget 6 which is near target 3 and has the same radius astarget 3 is not failed. In order to investigate the reason, wedeleted the failure criterion so that the rock will never fail(corresponding to an infinite rock strength). The re-calculation results in Fig. 12 show that the difference ofthe principal stresses between these two targets is verysmall. At the time of 8.5952 ms, s1 at target 3 is 112.8MPaand at target 6 is 112.4MPa. Because the principal stress attarget 3 is first reaching the tensile strength, 112.8MPa, theelement containing target 3 will fail first, and then its tensileand shear stresses will release, which will cause the tensilestress at target 6 to decrease. Thus, the element containingtarget 6 will not fail. If the element containing target 3 didnot fail first, from Fig. 12, s1 at target 6 would reach112.8MPa at the time of 8.5960 ms and the elementcontaining target 6 would fail.

4. Factors that influence crack initiation and propagation

Many factors influence rock fracturing, such as bound-ary conditions, coupling medium, borehole diameter,decoupling and joint. Based on the numerical model,EOS, and the failure criterion, the influences of thesefactors on crack initiation and propagation will beinvestigated in the follows.

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100

-100

-200

-300

-400

0

Str

ess

(MP

a)

0 2 4 6 8 10

Time (microsecond)

0.8853

σθσθ

σr

Fig. 7. Relationship of dynamic stresses (i.e. radial stress sr and tangential

stress sy) versus time for the element containing target 2

ðradius ¼ 12:114mmÞ.

Fig. 8. Material status and the stress wave front at the moment of 4.221ms after initiation.

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4.1. Boundary condition

Fig. 13 shows the material statuses of two 100mmdiameter rock samples with free and transmit boundary at

the time of 32.19 ms after detonation in the borehole. Wateris filled in the space between the explosive and the boreholewall. The free boundary represents a regular free surface inwhich stress wave will reflect back; the transmit boundarymeans that the stress wave transmits the boundary and isnot allowed to reflect back, which essentially is an analogof an infinitely large diameter rock sample. In the case offree boundary, reflected stress wave can cause circumfer-ential cracks. In contrast, the ‘transmit’ boundary case inFig. 13 at the same 32.19 ms time does not show any

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100

50

0

-50

-100

-150

Str

ess

(MP

a)

6 8 10 12

Time (microsecond)

σr

σr

σθ

σθ

τrθ

τrθ

Fig. 9. Relationship of dynamic stresses (i.e. radial stress sr, tangentialstress sy and shear stress try) versus time for the element containing target

4 ðradius ¼ 40:035mmÞ.

60

100

0

20

-20

Str

ess

(MP

a)

8 10 12 14

Time (microsecond)

σr

σθ

Fig. 10. Relationship of dynamic stresses (i.e. radial stress sr and

tangential stress sy) versus time for the element containing target 5

ðradius ¼ 50mmÞ.

Fig. 11. Sketch of the interaction gap between two contact materials.

Fig. 12. The curves of s1 versus time for the elements containing targets 3

and 6, respectively, obtained under the condition that the rock never fail.

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circumferential cracks due to the absence of reflected stresswave at the boundary, as expected.

The above discussion is based on a relatively smalldiameter rock sample (100mm in diameter). Fig. 14 showsthe material statuses of two 300mm diameter rock sampleswith free and transmit boundary at 61.5 ms after detona-tion. The reason of the longer time duration (61.5 ms) isbecause the longer time will take for the stress wavereflecting back from the outer boundary to the borehole.For the case of free boundary, the maximum crack length is108.0mm, whereas, for the case of transmit boundary it isonly 61.2mm. This represents a 76.5% increase in terms of

the maximum crack length when the transmit boundary isreplaced by free boundary. This indicates that using freeboundary (or free surface) in blasting, the extent offragmentation will increase significantly, which is inagreement with practice. It can be seen that no circumfer-ential cracks occur in the case of free boundary. This isbecause after the attenuation of the stress wave’s magni-tude in a longer distance, the reflected stress waves are tooweak to cause circumferential cracks close to the freesurface. Actually the simulation results show that when thediameter of the current rock sample is larger than 240mm,spalling cracks will not occur.

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Fig. 13. Comparison of material statuses for the rock sample with free and transmit boundary; the rock sample outer diameter ¼ 100mm.

Fig. 14. Comparison of material statuses for the rock samples with free and transmit boundary; the rock sample outer diameter ¼ 300mm.

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4.2. Coupling medium

The objective of this investigation is to study shock orstress wave induced cracking in rock, and the fracturedrock mass movement driven by the explosion gases is notfocused. In order to reduce the effect of the subsequenthigh-pressure explosion gases on further development ofthe cracks, coupling material is applied and filled in theplace between explosive charge and borehole wall, which isalso a common practice in blasting.

Three types of materials, water, sand and air, are used ascoupling materials in this simulation. The correspondingmaterial statuses for three 300mm diameter rock sampleswith different coupling materials at 61.5 ms after detonationare shown in Fig. 15. The maximum crack length in thewater-coupled case is 108.0mm, whereas, for the sand-coupled and air-coupled cases they are 93.4 and 17.8mm,respectively. This is to be expected, as water is known toserve as an excellent medium for transmission of shock andstress wave, compared to sand or air. In the water-coupledcase, the maximum crack length is 6.1 times longer thanthat in the air-coupled case, whereas, the maximum cracklength when water is replaced by sand (or similar materials)decreases by about 13.5%. Thus, to minimize blast-induceddamage in rock, air-coupling of the explosive charge in aborehole would offer the best solution. This is in agreementwith current practice, which is based on more qualitativeand empirical approach.

4.3. Borehole diameter and decoupling

The effect of decoupling is studied by changing theborehole diameter from 10 to 6mm, while keeping the

same charge diameter of 1.08mm. The rock samplediameter is 400mm, and the coupling material is water inthis simulation. The corresponding material statuses at72 ms after detonation are shown in Fig. 16. For the rockand the explosive type concerned, the respective maximumcrack lengths are found to be 128.4mm ðD ¼ 6mmÞ and90.6mm ðD ¼ 10mmÞ. In terms of maximum crack length,this represents a 29.4% decrease when the boreholediameter (or decoupling) is increased from 6 to 10mm.In normal blasting practice, in order to decrease the

extent of fractures around the borehole the explosivecharge diameter is reduced while keeping the same bore-hole diameter throughout, as it is easier to change theformer than the latter. To simulate such conditions, threecharge diameters 0.54, 1.08 and 1.62mm are designed inthe same 10mm diameter borehole with water couplingmaterial. The corresponding material statuses at 72 ms afterdetonation are shown in Fig. 17. The maximum cracklengths are 29.5, 90.6 and 183.3mm, respectively. Theratios of the maximum crack length to the explosiveconsumption in weight (the sample thickness ¼ 1:0mm)are 107.4, 82.5 and 74.1mm/mg, respectively. It can be seenthat the ratio of the maximum crack length to explosiveconsumption decreases as the explosive charge diameterincreases.

4.4. Joint width and joint filling material

During the process of stress wave traveling, joints arefrequently encountered in a rock mass. The stress wavewould suffer both partial reflection and transmission acrossa joint plane. The transmission characteristics for the stresswave depend on joint spacing [19], joint orientation [20],

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Fig. 15. Comparison of material statuses for the rock samples with different coupling materials (water, sand and air); the rock sample outer diameter ¼

300mm and the outer boundary is free.

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joint width as well as the type of material filling these joint.The material statuses for the rock samples with differentjoint widths at the time of 23.4 ms after detonation areshown in Fig. 18. The diameter of the rock samples is130mm and the joint widths are 0.5, 1.0 and 1.5mm,respectively. Unconsolidated soil is taken to be the fillingmaterial of these joints. The extent of damage outside thejoint plane is reduced significantly, compared to the areawithout joint influence. As the joint width increases, the

extent of damage outside the joint plane decreases. For thecase of the joint width of 0.5mm, several radial cracks canpass through the joint plane. When the joint width isincreased to 1.0mm, the amplitude of the transmitted waveis sufficiently reduced, and only one crack can be observedpassing through the joint plane. When the joint width isincreased to 1.5mm, no cracks can pass through the jointplane. It can be seen that the density of the small cracks onthe incident side of the joint plane due to reflection of the

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Fig. 16. Comparison of material statuses for the rock samples with different borehole diameters (10 and 6mm) but the same explosive charge diameter; the

rock sample outer diameter ¼ 400mm.

Fig. 17. Comparison of material statuses for the rock samples with different explosive charge diameters (0.54, 1.08, 1.62mm) in the same 10mm diameter

borehole; the rock sample outer diameter ¼ 400mm.

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stress waves from the joint plane increases as the jointwidth increases.

If the joint filling material of soil is replaced by air, theamplitude of the transmitted stress waves across the jointplane is further reduced (Fig. 19). No cracks can passthrough the joint plane, and the damage extent outside thejoint plane decreases. However, there is an increase in thedensity of small cracks on the incident side of the jointplane, due to reflection of the stress waves from the jointplane.

5. Conclusions

This study emphasizes the role of stress wave loading oncrack initiation and propagation during the initial stage ofdetonation in a borehole. Actually, the final crack map andthe extent of damage to target rock will be the result ofcumulative action of the stress wave phase and thesubsequent explosion gas expansion phase, and would bemuch more extensive than those for the initial stageoutlined in this paper. However, the nature of all

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Fig. 18. Comparison of material statuses for the rock samples ðdiameter ¼ 130mmÞ with different joint thickness (0.5, 1.0, 1.5mm); filling material is soil.

Fig. 19. Comparison of material statuses for the rock samples ðdiameter ¼ 130mmÞ with joint, which is filled with different materials (soil and air).

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subsequent crack extension, branching and coalescencewould be largely governed by the initial crack patternsgenerated by the stress wave loading. These have veryimportant bearings on control and predictions of fractureand fragmentation of rock in actual blasting.

The numerical model presented in this paper can beapplied to treat the problem of blasting-induced crackinitiation and propagation in rocks. Three basic fracturezones, i.e., crushed zone, severely fractured zone andincipiently cracked zone, around a borehole in the blastingprocess as well as the circumferential spalling cracks havebeen successfully simulated. The results show that shearstress (resulting from intense compressive stress) causes thecrushed zone in the immediate vicinity of the borehole, themajor tensile principal stress causes the radial cracks, andthe reflected stress causes the circumferential spalling cracks.

Boundary conditions affect rock fracturing. For thesample with free boundary (or free surface), the damageextent is significantly intensified, compared to those with-out free boundary. The maximum crack length for thesample with free boundary increases by more than 76%.Using coupling material filled in the space between theexplosive charge and the borehole wall can significantlyincrease the damage extent in rocks. In the water-coupledcase, the maximum crack length is more than six timeslonger than that in air-coupled case. If we keep the samecharge diameter, using small diameter borehole cansignificantly intensify the extent of rock fragmentation;and if we keep the same borehole diameter and change thecharge diameter, the ratio of the maximum crack length toexplosive consumption decreases as the explosive chargediameter increases. Joint reduces the amplitude of stresswave, and the damage extent outside the joint planedecreases as the joint width increases. If a joint is filled withsoil, the damage extent outside the joint plane willincreases, compared to those filled with air.

Acknowledgements

This work has been funded in part by the 973 project2002CB412700 and by NSFC-E090504.

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