A study on the behavior of rock mass subjected to blasting using modified distinct element method

To cite this paper: Int. J. Rock Mech. & Min. Sci. 34:3-4, paper No. 156. Copyright © 1997 Elsevier Science Ltd

Copyright © 1997 Elsevier Science Ltd

Int. J. Rock Mech. & Min. Sci. Vol. 34, No. 3-4, 1997 ISSN 0148-9062

To cite this paper: Int. J. RockMech. &Min. Sci. 34:3-4, Paper No. 156

A STUDY ON THE B E H A V I O R OF R O C K MASS S U B J E C T E D TO B L A S T I N G USING M O D I F I E D DISTINCT E L E M E N T

M E T H O D

M. K. K i m l ; S. E. Kim2; K. H. O h l ; W. J. K i m 1

1 School of Civil and Urban Engineering, Yonsei University, Seoul, 120-749, KOREA

2 Department of Civil Engineering, Sejong University, Seoul, 143-747, KOREA

A B S T R A C T

In this study, the behavior of rock mass subjected to blasting is investigated using a modified distinct element method. Rock mass is modeled by distinct elements connected by contact-area and contact-point springs. Contact-area springs are used for fully contacted area of continuum, and contact-point springs for partially contacted area for discontinuous faces. Blast loading is considered by the radial contact force surrounding the charge. The behavior of an interior point of charge is formulated by the conservative equations and the equation of state of ideal gas. The first order reaction rate and the thermal explosion initiation are applied. Lax-Wendroff Flux-Corrected Transport Algorithm is used to solve the governing equations. To investigate the effects of detonation wave and gas pressure on the behavior of rock mass, two loading cases corresponding to loading shape and three loading cases to blasting energy are studied. The gas pressurization and stress wave mechanisms are observed from the analysis using the MDEM.

Copyright © 1997 E l s e v i e r Sc i ence Ltd

K E Y W O R D S

Distinct Element Method • Modif ied Distinct Element Method • Numerical Simulation • Discontinuous Faces • Fragmentation • Contact Spring • Thermal Explosion • Blast Load • Colliding and Rebounding Behavior

I N T R O D U C T I O N

The phenomenology of explosive effects in rock mass has been of interest to researchers and engineers (Starfield et al. 1968). Recently, various numerical methods are developed to analyze the behavior of rock mass subjected to blasting (Swoboda et al. 1988). Heuz6 et al. 1993 used discrete elements for hard rocks under explosive loading. They simulated the discontinuities of rock mass and traced the large motions, but not the colliding and rebounding of rock mass for complete simulation of explosion effects.

Cundall et al. 1979 proposed a distinct element method for describing the mechanical behavior of assemblies of discs. Antonellini et al. 1995 formulated the stiffness of distinct element using Hertz theory for colliding and Mindlin's solution for slipping. However, the distinct element method was used only for granular medium.

On the other hand, Hakuno et al. 1989 developed the modified distinct element method (MDEM). The MDEM was used to simulate the behavior of the discrete, composite, and continuous media. However,

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the colliding and rebounding behavior was not simulated accurately because only the linear elastic element spring was used.

In this study, an MDEM is improved. Rock mass is modeled by distinct elements connected by contact-area and contact-point springs. The former is for predicting the behavior of the continuum, and the latter is for the behavior of discontinuous faces, such as colliding, rebounding, and slipping. A blast loading is calculated with the help of the conservative equations and the equation of state of ideal gas. The first order reaction rate and the initiation of thermal explosion are applied. The multi-steps of reaction processes of the blasting are simplified as a single step. Lax-Wendroff Flux-Corrected Transport Algorithm is used for solving the governing equations.

M O D E L I N G O F R O C K M A S S

Rock mass is categorized into the intact and the jointed rock. In this study, modified distinct elements are used for modeling the rock mass. The two types of contact spring are modeled: contact-area spring for continuum; contact-point spring for discontinuous faces.

Continuum

The fully contact face of intact rock in Figure 1 (a) is idealized by the contact area between elements in Figure 1 (b). The force is assumed to transmit through the contact area shown in Figure 1 (c). The stiffness of normal and shear springs can be determined by Eqn. 1, and Eqn. 2, respectively.

E A x - ( I )

n L

GA K - (2 )

s L

where subscripts n and s indicate normal and shear direction to the contact face, respectively. K is the stiffness of spring, E is the elastic modulus of the mass, and G is the shear modulus.

Tensile and shear failure criteria are described by Eqn. 3 and Eqn. 4, respectively.

(F)ouow = o , " A ( 3 )

(F).llo,~ = F t a n ~ + c " A (4)

where F n is the normal force, F s is the shear force, (Yt is the tensile strength, A is the contact area, ~) is the

internal frictional angle, and c is the cohesion.

Discontinuous Faces

The jointed rock mass includes the partially contact faces in Figure 2(a). Normal and shear forces on the partially contact faces cause interactions including colliding and slipping.

The discontinuous faces can be modeled by contact-point spring of Antonellini et al. 1994. In colliding, the stiffness of normal spring and of shear spring can be determined by Eqn.5 and Eqn. 6.

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K n

~G

(5)

r~G6 K = " (6)

$ r ( l -v )

where K is the stiffness of the contact-point spring, G is the shear modulus, 5 n is the overlapping level of

two elements, and r is the effective radius of the elements in contact. The stiffness depends on the overlapping level.

The sliding occurs at the contact point when the shear force exceeds the allowable frictional force. The allowable frictional force is determined by the Coulomb failure criterion (Eqn. 7).

(F)~Ho,~ = F tang (7)

where F n is the normal force, F s is the shear force, and ~ is the internal frictional angle. The shear contact

force keeps the allowable frictional force when sliding occurs.

MODELING OF BLAST LOADING

Blasting reaction starts at the ignited point, and propagates to the adjacent portion along the charge. In the progressive reaction, a large amount of chemical energy is released, and a shock wave occurs. The blasting phenomena are formulated by the conservative equations and the equation of state of ideal gas. Herein, the blast loading will be modeled using the radial contact force.

Governing Equations The behavior of an interior point of charge is described by the conservative equations for mass, momentum, and energy as Eqn. 8, Eqn. 9, and Eqn. 10.

0p + 0(pu) :0 (8) Ot c3x

O(pu) + c3Jpu 2 +p) =0 (9) Ot Ox

0(pe,)l l ~ O(pue'+pU).o (10) Ot Ox

where p is the density, u is the velocity, and p is the pressure, e t is the total energy consisting of the

internal energy, the kinetic energy, and the chemical reaction energy (Eqn. 11). These equations are similar to the mathematical model of one-dimensional hydrodynamics.

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] 2 e , =e + - - u + q z ( 1 1 ) 2

where e is the internal energy, q is the heat release, and z is the mass fraction. The behavior of gas products is assumed to be described by the equation of state for ideal gas (Eqn. 12).

1 e - p V (12)

y - 1

where Vis the volume and 5' is the specific heat ratio.

The reaction energy (qz) is added to the total energy (et) corresponding to the mass fraction. The

multi-steps of the reaction process of the blasting are simplified as a single step. The mass fraction is determined by the first order reaction rate model of Eqn. 13.

T) (13) g = Zo e

where k is the pre-exponential factor of Arrhenius equation, t is the time, T~. is the critical temperature

and H is the heavyside step function. A simple initiation condition is assumed, which is based on the thermal explosion initiation by Campbell et al. 1961. The charge is ignited when it has higher temperature than the critical one at least for the induction time.

Blast Loading

In blasting, a discontinuous wave occurs. In this study, Lax-Wendroff Flux-Corrected Transport Algorithm suggested by Book 1981 is used for capturing the discontinuous wave. The pressure at arbitrary position along the charge is determined by solving the governing equations. Herein, blast loading being a function of radius and length of the charge is modeled by the radial contact force determined by the pressure history. The force is loaded to surrounding media as shown in Figure 3(a). In non-crack state, gas products are confined by elements, and the blast force is transmitted to the nearest six elements shown in Figure 3(b). In cracked state, the pressure is distributed to the next surrounding six elements through the gap between two nearest elements shown in Figure 3(c). As a result, magnitude of blast force can be calculated by Eqn. 14 and Eqn. 15.

Fbtastmg = p r e s s u r e " A • t~ (i4)

Where

0~ =

1

A

A +gap

gab

A +gap

: for uncrack state

: for c rack state to six neares t element.~

: f o r crack state to six near e lements

(15)

where ct is a factor for pressure distribution, A is the contact area, and gap is the clear distance between

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two nearest elements.

Volumetric expansion of a borehole due to blast loading is described using the equation of state of ideal gas (Eqn. 16).

2 Do

p - D e

x Po (16)

wherep is the pressure in the volumetric expansion state, Po is the pressure in the perfectly confined

state, D is the distance between the center of blast loading and the contact point of the nearest element, and D o is the initial radius of the borehole.

V E R I F I C A T I O N

Compression Strength

To verify the failure condition of the contact spring, the behavior of a rectangular specimen subjected to the static compressive loading is investigated. The configuration of the specimen and material properties are shown in Figure 4(a) and its distinct element model is shown in Figure 4(b). An incremental force is applied on the top of the specimen. When the compressive force reaches 2.297MN, the failure occurs. The compression strength obtained by analysis is 19.14MPa, which is slightly lower than 20.6MPa (7.1%) obtained by compression test. This difference seems to be caused by the loading condition. If the realistic loading algorithm is designed, the accurate result can be obtained. As a result, the MDEM is adequate to predict the compression test.

Blast Loading

The blast loading of trinitrotoluene (TNT) is simulated with the properties of TNT shown in Table. 1.

The pressure profile of blasting as a function of time and charge length is shown in Figure 5. A blast wave occurs when the charge is ignited, and propagates along the unreacted charge rapidly. The pressure following the Von-Neumann spike tends to approach the Chapman-Jouguet (C-J) pressure. The pressure and the detonation velocity at C-J state predicted by the proposed method are compared with the results of Cowan et al. 1956 in Table 2. The detonation velocity of the blast loading is lower than of Cowan et

a l . . However, the pressure of the blast loading is higher than of Cowan et a l . . The proposed formulation can show clearly the blasting pressure profile including; Von-Neumann spike, C-J point, and Taylor wave form. The realistic parameters for the model should be studied in the future, since the absolute value are slightly different from Cowan et a l . .

N U M E R I C A L ANALYSIS

Simulation of the Behavior of Rock Mass subjected to Blasting

Behavior of intact rock including a cylindrical borehole is investigated using the improved MDEM. Figure 6 shows the intact rock mass and its MDEM model. Two loading shapes with 0.02p are considered shown in Figure 7, in which p is the calculated pressure for TNT of 10cm in length and lcm in radius. Three loading cases are considered corresponding to blasting energy: 1.0p; 0.1p; and 0.02p.

Figure 8 shows the rock fracture patterns corresponding to loading shapes: blast loading including gas

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pressure; and shock loading not including gas pressure. Near the borehole, the cracks due to shear force is observed in blast loading case, which is the effect of gas pressure. The cracks due to tensile force is observed in shock loading case, which is the effect of the nonexistance of gas pressure. However, the failure patterns near the surface are similar in both cases. As a result, the gas pressure affects the behavior of rock mass near the borehole, while it does not affect the behavior of the domain where the stress wave mechanism appears.

Figure 9 shows the rock fracture patterns corresponding to blast energy. The crack region becomes wider with the larger energy. Shear failure is more dominant on the frictional materials in compressive loading than tensile failure. The gas pressurization mechanism is shown to govern the failure of frictional material near borehole.

In Figure 8(a) and Figure 9(c) with the same blasting loading, it is investigated that the cracks due to tensile force increase with time, while the cracks due to shear force are not changed. It seems that the stress wave reaches the surface as compressive wave, and is reflected as tensile force causing the cracks. This crack propagation process shows the stress wave mechanism explained by Fourney 1993.

C O N C L U S I O N

An MDEM is improved to simulate the behavior of rock mass subjected to blasting. Rock mass is modeled by the distinct elements connected by the contact-area and the contact-point springs. The blast loading is described by the contact force surrounding the charge. To investigate the effects of stress wave and gas pressure on the behavior of rock mass, two loading cases corresponding to loading shape and three loading cases to blasting energy are considered. The gas pressurization and the stress wave mechanisms are observed from the analysis using the MDEM. The energy required in the gas pressurization mechanism is much greater than the energy required in the stress wave mechanism. The gas pressure affects the behavior of rock mass near the borehole, while it does not affect the behavior of the domain where the stress wave mechanism appears. Both stress wave and gas pressure are important in the fracture and fragmentation of rock mass. It is concluded that the MDEM can be used for a practical tool simulating the rock blasting.

Acknowledgments This paper was supported by NON DIRECTED RESEARCH FUND, Korea Research Foundation.

FIGURES

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To cite this paper: Int. J. Rock Mech. & Mm. Sci. 34:3-4, paper No. 156. Copyr ight © 1997 Elsevier Science Ltd

Paper 156, Figure 1.

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To cite this paper: Int. J. Rock Mech. & Mm. Sci. 34:3-4, paper No. 156. Copyright © 1997 Elsevier Science Ltd


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ISSN 0148-9062

To cite this paper: Int. J. Rock Mech. & Mm. Sci. 34:3-4, paper No. 156. Copyright © 1997 Elsevier Science Ltd


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To cite this paper: Int. J. Rock Mech. & Mm. Sci. 34:3-4, paper No. 156. Copyr ight © 1997 Elsevier Science Ltd


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" i i i i " " i • i

I I

( ' l ' , ' l ! . ~k ( - ; ...':1.1...: I~'.:. ~h¢ .n r I , b r . z c

, - , , : - ¢ % , , , . . . - j . r - 7 - ~ - - -

: . . . : . . . . ~ .J . l I

c~ :s.: ...... c~., : " ' " I . " " " I I I "" "

i

Cracks ,:[uc Ld-..:hcizr I'tm.'c

i

." . ~'" t ' l

L'" , - . O

Figure 9. Rock fracture patterns corresponding to blast energy (at t=40psec)

TABLES

ISSN 0148-9062


Paper 156, TABLE 1.

TABLE 1 P R O P E R T I E S O F E X P L O S I V E T N T

. .. Property Heat Release

Pre-exponential Factor i i i

Critical Temperature Induction Time

Symbol q V

T0 tind

Unit ca'i)g

C sec

Value 1000

1.0x 10 t4 300 0.27

Reference Kim, J. K., 1986

Hubbard, H.W. et al., 1959 Kim, J. K., 1986

Paper 156, TABLE 2.

TABLE 2 DETONATION VELOCITY AND PRESSURE AT C-J STATE

" Blast Loadi'ng

] Detonation Veloc!ty (m/,sec) Pressure (GPa) 5714 22.1

i m l ' '

Cowan (Experiment) 6840 16.8 C owan (Calculation) " 6894 19.6' . ....

References

References

Antonellini M. A., Pollard D. D. 1995. Distinct Element Modelling &Deformation Bands in Sandstone. Journal of Structural Geology, 17:8, 1165-1182.

Book D. L. 1981. Finite-DifJbrence Techniques for Vectorized Fluid Dynamics Calculations, Springer-Verlag New York Inc.

Campbell A. W., Davis W. C, Travis J. R. 1961. Shock Initiation of Detonation in Liquid Explosives. The Physics of FIuids, 4:4, 498-510.

Cowan R. D., Fickett W. 1956. Calculation of the Detonation Properties of Solid Explosives with the Kistiakowsky-Wilson Equation of State. The Journal of Chemical Physics, 24:5, 932-939.

Cundall P. A., Strack O. D. L. 1979. A Discrete Numerical Model for Granular Assemblies. Geotechnique, 29:1, 47-65.

Fourney W. L. 1993. Mechanisms of Rock Fragmentation by Blasting. Comprehensive Rock Engineering:

ISSN 0148-9062


Principles, Practice, and Projects, editor-in-chief Hudson, J. A., 1st ed., 4-1, Pergamon Press, 39-69.

Hakuno M., Meguro K. 1989. Fracture Analysis of Concrete Structures by the Modified Distinct Element Method. Journal of Slructural Engineering and Earthquake Engineering, Japan Society of Civil Engineers, 6:2, 283s-284s.

Heuz6 F. E., Walton O. R., Maddix D. M., Shaffer R. J., Butkovich T. R. 1993. Analysis of Explosions in Hard Rocks: The Power of Discrete Element. Comprehensive Rock Engineering: Principles, Practice, and Projects, editor-in-chief Hudson, J. A., 1st ed., 2-1, Pergamon Press, 387-413.

Hubbard H. W., Johnson M. H. 1959. Initiation of Detonations. Journal of Applied Physics, 30:5, 765-769.

Kim J. K. 1986. Explosives and Blasting Engineering, Seoul National University Press, Seoul, Korea.

Persson R A., Holmberg R., Lee J. M. 1994. RockBIasting andExpIosives Engineering, CRC Press Inc.

Starfield A. M., Pugliese J. M. 1968. Compression Waves Generated in Rock by Cylindrical Explosive Charges: A Comparison between a Computer Model and Field Measurements. International Journal of Rock Mechanics and Mining Science, 5, 65-77.

Swoboda G., Zenz G. 1988. Practical Considerations of Blast Loading in Tunneling Using Numerical Modeling. Numerical Methods in Geomechanics, Edited by Swoboda, G., Balkema, Rotterdam, 1681-1688.

ISSN 0148-9062

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A study on the behavior of rock mass subjected to blasting using modified distinct element method