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Modelling of blast-induced fractures in jointed rock masses

Z.L. Wang a,b,*, H. Konietzky b

a Department of Modern Mechanics, University of Science and Technology of China, Hefei 230027, Chinab Institut für Geotechnik, Technische Universität Bergakademie Freiberg, Freiberg 09596, Germany

a r t i c l e i n f o

Article history:

Received 25 August 2008

Received in revised form 16 March 2009

Accepted 9 May 2009

Available online 18 May 2009

Keywords:

Jointed rock masses

Blasting

Coupled method

Dynamic fracture

Numerical modelling

a b s t r a c t

This paper focuses on the dynamic fracturing process of jointed rock masses due to blastwave loading. A coupled numerical method using both LS-DYNA (a transient dynamic finite

element program) and UDEC (an universal discrete element code) is outlined first. The

blast-induced fracture extension in two representative jointed rock masses is studied sub-

sequently. Besides, the effects of loading density of explosive charge on fracture evolution

and the effects of the preexisting earth stress and free face on the fracturing process of rock

masses are also explored in detail. The numerical results capture some of the well-known

phenomena observed in the field and expected theoretically, and thus can offer useful

guidelines to blasting design in rock masses.

2009 Elsevier Ltd. All rights reserved.

1. Introduction

The destruction of hard rocks by means of blasting usually involves the drilling of a borehole, placement of an explosive

charge and stemming prior to detonation. When the explosive is detonated, an extremely high pressure pulse is generated

which is transmitted into rock mass adjacent to the borehole, producing a dilatational wave that propagates away from the

charge. This may cause damage to rock and, furthermore, when the compressive stress wave reaches a free face or fissure

(non-transmission), it will be reflected and converted into tensile wave, which may produce tensile cracking or cause spall-

ing of surficial slabs if the tensile strength of the rock is exceeded [1–3].

Some researchers [4,5] believe that cracking is mainly caused by the incident dilatational wave and any reflected waves,

while other investigators [6] consider the action of the compressed gases forcing its way through the cracks from the bore-

hole more important. Until recently, it is generally agreed that both stress wave and gas pressure loadings play an important

role in the process of rock fracture and fragmentation. In fact, our understanding of the blasting process is far from thorough,

as both the explosive and the rock are complex materials. The release characteristics of energy stored in the explosive charge

are highly variable, depending on the predominant factors like borehole parameters and charge loading density. Similarly,the response of the circumambient rock mass to high strain-rate dynamic loadings, which may last only for a few millisec-

onds, remains largely unknown. Under this scenario, the best approach to the study may be to first generate an extensive

experimental database on these properties. At the same time, it is necessary to explore the fracture and fragmentation pro-

cesses through numerical tools so as to obtain a better understanding of the underlying mechanism [3,7].

It is well recognized that the properties of a rock mass are determined by the properties of the intact rock and the dis-

continuities such as joint structures and faults [8,9]. The presence of the discontinuities has significant influence on the re-

sponses of the rock mass to either static or dynamic loading, and renders the numerical simulations more complicated [9,10].

0013-7944/$ - see front matter 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.engfracmech.2009.05.004

* Corresponding author. Address: Department of Modern Mechanics, University of Science and Technology of China, Hefei 230027, China. Fax: +86 551

3606459.

E-mail address: [email protected] (Z.L. Wang).

Engineering Fracture Mechanics 76 (2009) 1945–1955

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There are some numerical codes available for simulating discontinuous rock mass at present, the popular ones being the fi-

nite element method (FEM), boundary element method (BEM), finite difference method (FDM), and discrete element method

(DEM). For the joints, they can be modelled as a kind of special ‘‘interface elements’’ in FEM [11], or simulated using ‘‘slide

lines” in FDM, etc [12]. However, these numerical treatments are incapable of dealing with many intersecting interfaces and

have no an automatic scheme for recognizing new contacts. In addition, they are limited to small displacements and/or rota-

tion, etc. UDEC, a two-dimensional discrete element code, is proven as an ideal numerical technique for modelling the joint-

or fault-related problem in discontinuous rock system [13,14]. The rock mass is represented as an assemblage of discrete

blocks, and the internal discontinuities are treated as boundary conditions between blocks. Large displacements along dis-

continuities and rotations of blocks are allowed. Individual blocks behave as either rigid or deformable materials. Deform-

able blocks are subdivided into constant-strain elements or zones, and each zone follows a prescribed stress-strain law. In

addition, an appropriate material model must also be assigned to all discontinuities in calculations. Generally, the area con-

tact Coulomb slip model is the most commonly used model for engineering studies [14].

This study concentrates on the role of stress-wave induced fracture in jointed rock masses because the rock fracturing

process under stress wave loading is considered the crucial stage as the subsequent rock fragmentation and large-scale

movement due to gas pressure loading [7]. After the coupled method combining LS-DYNA and UDEC is introduced in detail,

the effects of joint structure, loading density, earth stress as well as free face on the fracture creation and evolution are ex-

plored by assuming that the blocks behave elastically and the joints follow the Coulomb slip behavior.

2. Numerical simulation tools

In the present study, the numerical analyses are carried out using a coupled method which combines the LS-DYNA and

the UDEC together. The former generates the blast loading which is used by the latter to simulate the blast-wave propagation

in jointed rock mass.

LS-DYNA is a general-purpose commercial code, which can be applied to solve a wide variety of non-linear problems in

solid, fluid and gas dynamics [15]. This code has been proved by numerous studies as a suitable tool for large deformation

analysis including the rock fracturing [3,16]. It performs dynamic analysis by seeking a solution to the momentum equation,

which satisfies all boundary conditions, while integrating the energy equation to be used as a balance for global energy. Kine-

matic boundary conditions are applied when the equations of motion and constitutive equations are solved. Time integration

is conducted via the central difference method when solving for displacements, velocities and accelerations. The timestep is

Nomenclature

Drn normal stress incrementDun normal displacement incrementkn normal stiffnessks shear stiffnessrn normal stress

ss shear stressrt limiting tensile strengthsmax maximum shear stressC joint cohesion

/ joint friction anglew joint dilation angleDues elastic incremental shear displacementDus total incremental shear displacementucs critical shear displacementql loading density of chargeq0 mass density of rockE young’s modulus of rockv poisson’s ratio of rockC

1, r

1, C

2, r

2, x

constants of explosiveP pressureV specific volumee specific energye0 initial specific energyk wavelengthDl spatial element sizet timeS h horizontal effective stressS v vertical effective stress

1946 Z.L. Wang, H. Konietzky / Engineering Fracture Mechanics 76 (2009) 1945–1955

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determined by the smallest element in the entire model. Databases are then generated which record history variables such

as stress, strain and kinetic energy. Finally, LS-DYNA updates the current time and checks it against the simulation termina-

tion time.

It has been well known that handling discontinuities in rock masses is a challenging task. UDEC is specially developed to

model discontinuous problems [14]. It can accommodate a large number of discontinuities and permits the modelling sys-

tem undergoing larger geometrical change through the use of contact updating scheme. In UDEC, the deformation of a frac-

tured rock mass consists of the elastic/plastic deformation of blocks of intact rock, together with the displacements along and

across fractures. The motion of block is characterized by the Newton’s second law of motion, which is expressed in the cen-

tral finite difference form with respect to time. Calculations are performed over one timestep in an explicit time-marching

algorithm. For deformable blocks, numerical integration of the differential equation of motion is used to determine the

incremental displacements at the gridpoints of the triangular constant strain element within the blocks. The incremental

displacements are then used to calculate the new stresses within the element through an appropriate constitutive equation.

The amount of normal and tangential displacement between two adjacent blocks can be determined directly from

the block geometry and the translation and rotation of the centroid of each block, as will be described in the upcoming

sections.

3. Coupled method and joint model

In this study, the coupled method combining LS-DYNA and UDEC is employed to simulate the blast-wave propagation and

associated dynamic fracturing in jointed rock masses. An illustration of the above-mentioned coupled method (refer to

Fig. 1) is elaborated as follows:

Firstly, LS-DYNA is used to model the explosion process of the explosive whereby the rock mass is temporarily assumed as

a continuous medium. The explosion history is measured on the wall of the borehole (or chamber) in terms of particle

velocity or pressure (see Fig. 1b). Secondly, after the explosion history is converted into the formatted data compatible with

UDEC, it is fed into the UDEC modelling as a radial velocity history. Thirdly, the blast-wave propagation and fracture evolu-

tion in the jointed rock mass, induced by detonating the explosive charge, is simulated using the UDEC modelling (see

Fig. 1c).

The intact rock blocks are assumed to be perfectly elastic in the present study. As for the joint behavior model, many types

of constitutive laws may be contemplated. It has been proven that the Coulomb slip model with residual strength (area con-

tact) is capable of capturing several of the features which are representative of the physical response of joints [14] and is thus

adopted in the present study.

In the normal direction, the stress-displacement relation is assumed to be linear and governed by the stiffness kn (ex-

pressed in terms of incremental stress per unit of displacement) as

Drn ¼ knDun ð1Þ

where Drn is the normal stress increment of the interface between blocks, Dun is the normal displacement increment.Also, there is a limiting tensile strength, rt , for the joint. If the tensile strength is exceeded, i.e.,

rn < rt ð2Þ

then let rn = 0.Similarly, in the shear direction, the response is controlled by a constant interface shear stiffness, ks. The shear stress ss, is

limited by a combination of joint cohesion (C ) and friction angle (/). Thus, if

ss 6 smax ¼ C þ rn tan/ ð3Þ

then,

Dss ¼ ksDues ð4Þ

or else, if

ss P smax ð5Þ

then,

ss ¼ signðDusÞsmax ð6Þ

where Dues is the elastic component of the incremental shear displacement, Dus is the total incremental shear displacement.

In addition, joint dilation may occur at the onset of slip of the joint, the dilation is restricted such that [14]:

w ¼ 0 if ss 6 smaxif ss ¼ smax and jusjP ucs

ð7Þ

where w is the dilation angle, ucs denotes the critical shear displacement.

Z.L. Wang, H. Konietzky / Engineering Fracture Mechanics 76 (2009) 1945–1955 1947


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Based on this joint model, a fracture is considered to be initiated when the joint between two elements exceeds its tensile

or shear strength, and the fracture begins to propagate when new discontinuities are created at its tip.

Fig. 1. Illustration of the coupled method.



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Fig. 3 plots the blast loading ( x-velocity at the borehole wall) computed by LS-DYNA. Due to this measured point( x = 0.3 m, y = 0 m) as marked in Fig. 2a on the x-axis, the obtained explosion history can be regarded as the radial velocity.

On the other hand, since the explosion history from the LS-DYNA is obtained randomly with time, it must be converted into a

compatible form (see, for example, equal time-spacing) before feeding to the gridpoints on the borehole boundary in the

UDEC modelling (see Fig. 2b).

The damping in the numerical simulations should attempt to reproduce the energy loss in the system when subjected to

dynamic loading [14]. Rayleigh damping is commonly used to provide damping that is approximately frequency-indepen-

dent over a restricted range of frequencies. In order to find the principal frequency of the input velocity history (refer to

Fig. 3), a spectral analysis should be made before simulations. As can be seen from Fig. 4, the principal frequency of the blast

wave (about 1657.0 Hz) is obtained via fast Fourier transform algorithm.

Table 2

JWL Parameters for TNT explosive.

q0 (kg/m3) C 1 (GPa) C 2 (GPa) DC – J (m/s) r 1 r 2 x e0 (GPa)

1640.0 373.77 3.23 6930.0 4.15 0.95 0.30 7.0

DC – J , the C–J detonation velocity.

0 2 4 6 8 10-5

0

5

10

15

20

Time (ms)

V e l o c i t y ( m / s )

Fig. 3. Explosion history calculated by LS-DYNA (ql = 410.0 kg/m3).

0 500 1000 1500 2000 2500 3000 35000

0.05

0.1

0.15

0.2

Frequency (Hz)

F o u r i e r a m p l i t u d e

Principal frequency

Fig. 4. Power spectrum of input velocity–time history.



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Fig. 9 shows the effect of loading density on the fracture pattern of rock mass at the same time (t = 5 ms). A higher loading

density of charge (728.9 kg/m3) increases the number of fractures and causes the intense stress release around the running

fractures. The stress release induced by adjacent fractures disperses the concentrated stresses around the tips of the follow-

ing fractures and thus affects fracture propagation, resulting in many shorter factures [20]. When the loading density de-

creases to lower level (410.0 kg/m3), the fracturing due to tension and shear loads markedly reduces and there are

comparatively longer and continuous fracture extension (see Fig. 9b). When the loading density of charge decreases further

to 182.2 kg/m3, only fewer fractures as shown in Fig. 9a can be observed.

4.3. Effect of preexisting stress field

In deep underground operations, blasting can be affected by earth stress. High earth stress field may influence dynamic

fracture process and orientation around the borehole. For example, the rock fractures may be extended or suppressed by pre-

existing stress field. In general, the high earth stress field can induce stress concentrations around the wall of the borehole

and further lead to non-uniformity in the fracture pattern.

In the present numerical simulations, two opposite edges of the numerical modelling (ql = 728.9 kg/m3) are pre-loaded

with three different stress fields (see Fig. 10). The minor horizontal effective stress S h is set equal to 0.3 MPa, acting parallel

to the x-axis. The major vertical effective stress S v is set equal to 10MP, 30 MPa or 50 MPa for different models acting parallel

(a) t t =0.5ms (b) =1.0ms (c) t =5.0ms

Fig. 6. Fracture evolution in Voronoi jointed rock mass.

(a) t t =0.5ms (b) =1.0ms (c) t =5.0ms

Fig. 7. Fracture evolution in orthogonally jointed rock mass.



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0 2 4 6 8 10-15

-10

-5

0

5

10

15

20

25

Time (ms)


(b)l

ρ =728.9kg/m3

0 2 4 6 8 10-2

0

2

4

6

8

Time (ms)


(a)l

ρ =182.2kg/m3

Fig. 8. Explosion histories of different loading densities.

(a)l

ρ =182.2 kg/m3l

ρ 3

(b) (c)l

ρ =410.0 kg/m =728.9 kg/m3

Fig. 9. Effect of loading density on rock fracturing (t = 5.0 ms).



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to the y-axis. Fig. 10 clearly demonstrates that the blast-induced fractures are aligned in the direction of the principal stress

axis (vertical), while the crack extension in the horizontal direction is intensively suppressed under a higher S v. The results

are in good agreement with some reported numerical and experimental observations in the literature [16,21].

4.4. Effect of nearby free face

In practice, rock breakage events caused by applied borehole pressures are usually performed near free boundaries to in-

crease the fracturing effect on rock masses. It is well known that when a compressive stress-wave reaches the free face of a

rock mass, it will be reflected backward to produce a tensile wave. If the reflected tensile wave is sufficiently strong (higher

than the dynamic tensile strength of rock), it may cause cracking and even spalling of surficial slabs [22].

Fig. 11 shows the effect of free face on rock fracturing at t = 2.0 ms. Three different numerical models with one free face or

two free faces are presented. It can be noticed that fractures are first initiated and extend from the borehole surface without

directional preference. With further propagation of the stress waves, the fractures caused by the tensile wave starts to ap-

pear near the non-transmission or reflecting boundary and are approximately parallel to the boundary. Some fractures inter-

lace together and are extended to a considerable length during the spalling.

(a)hS =0.3Mpa, vS hS =0.3Mpa, vS=10MPa (b) =30MPa (c) hS =0.3Mpa, vS =50MPa

Fig. 10. Effect of stress field on rock fracturing (t = 1.0 ms).

(a) Upper free-face (b) Right free-face (c) Right and lower free-face

Fig. 11. Effect of free face on rock fracturing (t = 2.0 ms).



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5. Conclusions

The dynamic fracturing process in rock blasting has received considerable attention over the last few decades. This paper

uses the hybrid method of UDEC and LS-DYNA to model the underground explosion in jointed rock masses and to investigate

the influences of the dominant factors, including joint pattern, loading density, stress field and free face, on the rock fractur-

ing process. The following conclusions can be drawn from the present study:

(1) The coupled method is a good numerical tool for simulating blasting in jointed rock masses. LS-DYNA is adopted tomodel the explosive detonation and to provide the blast loading to be used in UDEC, which in turn simulates the

stress-wave propagation and rock fracturing process.

(2) Blasting in rock masses with different joint structures will behave differently. In a rock mass containing orthogonal

joints, the crack evolution exhibits strong anisotropy. The Voronoi joint generator is found to be very conducive for

numerical simulation of rock crack creation and propagation due to blasting. The fracturing happens when the joint

strength between Voronoi blocks is exceeded.

(3) The loading density of charge has a significant effect on the rock fracture pattern. A higher ql implies a higher peak of dilatational wave, which induces a larger fractured zone with many shorter cracks. However, there are only fewer frac-

tures when the loading density is lower.

(4) The existence of stress field considerably affects the fracture extension and causes the non-uniformity of rock fracture.

The fractures tend to evolve along the major stress axis. Besides, near the free face, the stress wave is reflected and

converted into tensile wave, resulting in the well-known phenomenon of surficial spalling.

(5) Because the discrete blocks of rock is considered as elastic media in our calculations, no plastic crushed zone can beobserved in the immediate vicinity of borehole. Therefore, there is still room for further refinement in future research

work.

Acknowledgements

This study was supported by the Program for New Century Excellent Talents in University (No. NCET-08-0525) the

Alexander von Humboldt Foundation, the Civil Aeronautics Joint Research Foundation (No. 60776821) and the Specialized

Research Fund for the Doctoral Program of Higher Education (No. 20070358073). The first author would like to thank

Dr. R.F. Shen for his embellishment of this paper.

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