a numerical study on wave transmission across multiple joints

ORIGINAL PAPER

A Numerical Study on Wave Transmission Across MultipleIntersecting Joint Sets in Rock Masses with UDEC

J. B. Zhu • X. F. Deng • X. B. Zhao • J. Zhao

Received: 23 March 2012 / Accepted: 8 December 2012 / Published online: 19 December 2012

� Springer-Verlag Wien 2012

Abstract This paper presents a numerical study on wave

transmission across jointed rock masses with UDEC, where

multiple intersecting joint sets exist. The capability of

UDEC of studying wave transmission across rock joints is

validated through comparison with analytical solutions and

experimental data. Through parametric studies on wave

transmission across jointed rock masses, it is found that

joint mechanical and spatial parameters including joint

normal and shear stiffnesses, nondimensional joint spacing,

joint spacing ratio, joint intersecting angle, incident angle,

and number of joint sets together determine the wave

transmission. And for P wave incidence, compared with

other parameters, joint normal stiffness, nondimensional

joint spacing, and joint intersecting angle have more sig-

nificant effects on wave transmission. The physical reasons

lying behind those phenomena are explained in detail.

Engineering applications and indications of the modeling

results are also mentioned.

Keywords UDEC � Wave transmission �Multiple intersecting joint sets � Parametric study

1 Introduction

The presence of joints is the most significant difference of

rock masses from other engineering materials. They are

often in the form of multiple intersecting joint sets. When a

wave transmits across jointed rock masses, its attenuation

and slowness are mainly induced by rock joints. In rock

engineering, peak particle velocity (PPV) is often adopted

as stability criteria of rock masses and rock structures

under dynamic and seismic loading. In addition, study of

wave transmission across jointed rock masses is also of

significant importance in geophysical investigation, earth-

quake seismology, and nondestructive evaluation.

Wave transmission across a single joint and a joint set

has been extensively studied with analytical and experi-

mental approaches (Morland 1977; Schoenberg 1980;

Pyrak-Nolte et al. 1990a, b; Watanabe and Sassa 1995;

Daehnke and Rossmanith 1997; Cai and Zhao 2000; Zhu

et al. 2011a; Perino et al. 2012). However, when multiple

intersecting joint sets exist, wave transmission is pro-

nouncedly complicated and hence, analytical solutions are

impossible to obtain and experimental tests are difficult to

conduct.

Numerical modeling is an economical and convenient

alternative to study wave transmission across jointed rock

masses. The representation of rock joints is a main diffi-

culty in numerical modeling. In the finite element method,

joints are often viewed as individual elements named joint

elements (Goodman et al. 1968; Ghaboussi et al. 1973).

Boundary interfaces are often adopted to represent joints

with the finite element method and boundary element

J. B. Zhu (&)

Graduate Aeronautical Laboratories and Department of

Mechanical and Civil Engineering, Division of Engineering

and Applied Science, California Institute of Technology,

Pasadena, CA 91125, USA

e-mail: [email protected]

X. F. Deng � J. Zhao

Ecole Polytechnique Federale de Lausanne (EPFL), School

of Architecture, Civil and Environmental Engineering (ENAC),

Laboratory for Rock Mechanics (LMR),

1015 Lausanne, Switzerland

X. B. Zhao

NJU-ECE Institute for Underground Space

and Geo-environment, School of Earth Sciences

and Engineering, Nanjing University, Nanjing 210093, China

123

Rock Mech Rock Eng (2013) 46:1429–1442

DOI 10.1007/s00603-012-0352-9

method (Beer 1986) or between boundary element methods

(Crotty and Wardle 1985). In the finite difference method,

joints are treated as slide lines (Schwer and Lindberg

1992). These treatments are applicable only when the

number of joints and their displacement are small. The

recently introduced distinct lattice spring model adopts

the virtual joint plane method to represent joints (Zhu et al.

2011b). However, the accuracy of the results is guaranteed

through adopting significantly small particle size, which

results in low computational speed and small model size. In

rock mechanics and rock engineering, due to its discon-

tinuous characteristics, the discrete element method

(DEM), which is capable of and has advantage in studying

discontinuous problems, has been widely used to study

problems related with jointed rock masses (Cundall 1971).

In DEM, a rock mass is represented as an assembly of

blocks and joints as interfaces between the blocks.

The universal distinct element code (UDEC), which is a

2D DEM-based numerical program, has been widely

adopted to study wave transmission across jointed rock

masses. In UDEC, blocks represent intact rock; zero-

thickness contacts represent joints, which are deformable

(i.e., the discontinuity is treated as a boundary condition).

Block movements are obtained from applied forces and

body forces with Newton’s second law. With a certain

force–displacement relation (joint model) at contacts/joints,

contact forces are obtained from joint displacements (clo-

sure, opening and slip), which are calculated from block

movements. The contact forces and displacements are

continuously found in a series of calculations, which trace

block movements and disturbance propagation caused by

applied loads and body forces. Lemos (1987) conducted a

UDEC modeling on S wave attenuation across a single joint

with Coulomb slip behaviour. Brady et al. (1990) performed

UDEC modeling on the slip of a single joint under an

explosive loading. Chen and Zhao (1998) modeled the

dynamic responses of a rock mass containing two inter-

secting joints under explosion loading with UDEC. The

effect of boundary conditions in UDEC on P wave trans-

mission across a single joint has also been studied (Fan and

Jiao 2004). Zhao et al. (2008) carried out numerical studies

of P wave transmission across a joint set with nonlinearly

deformational behavior with UDEC. Barla et al. (2010)

obtained numerical solutions for seismic wave propagation

through a rock column containing a joint set with UDEC.

However, no work has been conducted to systemically

study plane wave transmission across rock masses con-

taining multiple intersecting joint sets. Chen (1999) mod-

eled an underground explosion test in rock mass containing

multiple joint sets. Vorobiev (2010) and Vorobiev and

Antoun (2011) numerically studied cylindrical wave

propagation through two perpendicularly intersecting joint

sets.

In this paper, wave transmission across rock masses with

multiple joint sets is studied with UDEC. Through com-

parison with analytical solutions and experimental data,

effects of joint mechanical and spatial properties on wave

transmission in UDEC modeling are validated. Subse-

quently, parametrical studies on wave transmission across

rock masses containing multiple joint sets are conducted.

Seven parameters are adopted to describe a given jointed

rock mass. The dependence of wave transmission on those

parameters are discussed and explained in detail.

2 Verification of UDEC Modeling on Wave

Transmission Across Rock Joints

In order to verify the capability of UDEC on wave trans-

mission across rock joints, the modeling results are com-

pared with analytical solutions with the virtual wave source

method (Zhu et al. 2011a; Zhu and Zhao 2013) adopting the

displacement discontinuity model (Schoenberg 1980) and

experimental data in this section. Through comparison for

the case of normally incident P and S waves transmission

across a single joint, the effects of joint mechanical proper-

ties (joint normal and shear stiffness) and incident wave

types (P and S waves) on wave transmission in UDEC

modeling are verified. Effects of joint spatial configuration

and multiple wave reflections among joints on wave trans-

mission are subsequently verified through comparison with

theoretical solutions for normally incident P wave trans-

mission across a joint set. Obliquely incident P wave trans-

mission across a single joint is studied to verify the

dependence of wave transmission on the incident angle and

the two-dimensional effect. Finally, effects of multiple joint

sets on wave propagation in UDEC are validated through

comparison with an explosion field test in Alvdalen, Sweden.

Chen (1999) concluded that the mesh ratio defined as the

ratio of the mesh size to the wavelength must be smaller than

1/8–1/12 to ensure accurate modeling on wave transmission.

In this paper, to balance between accuracy and efficiency, the

mesh ratio is equal to 1/16. In addition, the deformational

behavior of joints is assumed to be linearly elastic both in the

normal and shear directions, and the rock material is assumed

to be elastic, isotropic, and homogeneous. It should be noted

that other verification studies on normally incident wave

propagation across a single and a joint set have also been

performed (Chen 1999; Zhao et al. 2008).

2.1 Normally Incident P Wave Transmission

Across a Single Joint

The UDEC model is illustrated in Fig. 1. The incident

wave is normally applied at the left boundary. The origin of

the X–Y coordinates is located at the left bottom of the

1430 J. B. Zhu et al.

123

model. Non-reflection viscous boundaries are placed at the

left and right boundaries to avoid wave reflections from the

artificial boundaries. The top and bottom side boundaries

are fixed in the y-direction. The joint is vertically located at

x = 2 m. Transmitted waves are measured at a point at

x = 2.4 m and y = 0.1 m. A one-cycle sinusoidal velocity

pulse with amplitude 0.1 m/s and frequency 1 kHz is

normally applied to the left boundary and propagates along

the x-direction through the model.

The material is assumed to be purely elastic. Although

UDEC is capable of considering energy loss due to mate-

rial inelasticity by additional damping parameter (there

exist two kinds of damping in UDEC, i.e., Rayleigh

damping and local damping), it is not included in this paper

for two reasons. First, when wave propagates across jointed

rock mass, the wave attenuation is mainly due to joints

(King et al. 1986; Zhao et al. 2006a). Second, this study is

focused on effects of rock joints on wave attenuation and

thus the attenuation from material is ignored. The assumed

basic properties of the rock material are as follows: the

rock density is 2,120 kg/m3, the Young’s modulus is

27.878 GPa/m, and the Poisson’s ratio is 0.2987.

The linear elastic joint model, termed as Coulomb slip

model, is the most general and popularly used one in

UDEC, and it is adopted in this paper. It is a joint area

contact model, which is intended for closely packed blocks

with area contact. The joint deformation/displacement is

represented by normal and shear stiffness in normal and

shear directions, respectively.

Figure 2 shows the magnitude of transmission coeffi-

cient across a single joint for P wave incidence T1Pj j as a

function of joint normal stiffness kn. It can be found that

T1Pj j from UDEC modeling agrees well with the analytical

solutions. Therefore, UDEC is applicable to study P wave

transmission.

2.2 Normally Incident S Wave Transmission


In order to verify the capability of UDEC in modeling S

wave transmission, normally incident S wave transmission

across a single joint is studied. The UDEC model, the

properties of rock material, and incident wave are the same

as those in Sect. 2.1, except that the incident wave is S

wave and the top and bottom side boundaries are fixed in

the x-direction.


cient across a single joint for S wave incidence T1Sj j as a

function of joint shear stiffness ks. It can be found that T1Sj jfrom UDEC modeling agrees well with the analytical

solutions. Therefore, UDEC is applicable to study S wave

transmission.

Fig. 1 The scheme of UDEC

model for P wave transmission

used for verification study

0 50 100 150 2000.0

0.2

0.4

0.6

0.8

1.0

lT1Pl

kn, GPa/m

Analytical solutions UDEC results

Fig. 2 T1Pj j versus kn for normally incident P wave transmission

across a single joint

0 50 100 150 2000.0

0.2

0.4

0.6

0.8

1.0

lT1Sl

Ks, GPa/m


Fig. 3 T1Sj j versus ks for normally incident S wave transmission


A Numerical Study on Wave Transmission 1431

123

2.3 Normally Incident P Wave Transmission

Across a Joint Set

When multiple parallel joints exist, wave transmission is

more complicated due to multiple wave reflections among

joints. The UDEC model, the properties of rock material,

and incident wave are the same as those in Sect. 2.1, except

that multiple joints exist with the rightmost joint located at

x = 2 m.


cient across a joint set for P wave incidence TNj j (N = 2, 5,

8) as a function of the nondimensional joint spacing n,

which is defined as the ratio of joint spacing to incident

wavelength, where kn ¼ 50 GPa/m. It can be found that

TNj j from UDEC modeling agrees well with the analytical

solutions. Therefore, UDEC is applicable to take into

account multiple wave reflections among joints.

2.4 Obliquely Incident P Wave Transmission


When the wave is obliquely incident upon a joint, wave

transformation as well as wave reflection and wave trans-

mission occurs. For a horizontally incident P wave across

an oblique joint, the transmitted S wave does not hori-

zontally propagate, although the transmitted P wave does,

as shown in Fig. 5. Therefore, wave reflection occurs at the

upper and lower boundaries, which will affect the trans-

mitted wave. In order to eliminate the effects of the

reflected wave from the upper and lower boundaries on the

transmission coefficient, the model size has been changed

to 30 m wide by 20 m high, and the transmitted wave is

measured at a point (x = 17 m, y = 10 m) just next to the

joint. By doing this, the reflected wave from the upper and

lower boundary cannot affect the transmission coefficient,

because the arriving time difference at the measuring point

between the transmitted wave across the joint and reflected

wave from the upper and lower boundaries is large enough

to separate them.

The UDEC model, the properties of rock material and

incident wave are the same as those in Sect. 2.1, except the

model size and the inclining joint. The incident angle (h) is

equal to the angle between the joint and the y axis.

In UDEC modeling, variables, e.g., velocity and dis-

placement, can be obtained in both x and y directions.

However, analytical solutions for obliquely incident wave

propagation across a single joint are in the form of coef-

ficients for transmitted P wave and transmitted S wave,

respectively. In order to compare the UDEC modeling

results and analytical solutions, analytical solutions for

transmitted wave in either x or y direction should be

obtained. The magnitude of transmitted wave in x direction

can be expressed by

0.0 0.1 0.2 0.3 0.4 0.50.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

lTNl

ξ

N=2 (Analytical solutions) N=2 (UDEC results) N=5 (Analytical solutions) N=5 (UDEC results) N=8 (Analytical solutions) N=8 (UDEC results)

Fig. 4 TNj j versus n for normally incident P wave transmission

across a joint set for different number of joints

Fig. 5 Obliquely incident wave

reflection, transmission, and

transformation on the joint,

where h is the angle of incident,

reflected, and transmitted P

wave, / is the angle of reflected

and transmitted S wave


123

WxðX; tÞj j ¼ ½ðWPðX; tÞ sin h�WSðX; tÞ cos /Þ2

þ ð�WPðX; tÞ cos h�WSðX; tÞ sin /Þ2�1=2;

ð1Þ

where Wx is the transmitted wave in x direction, WP and WS

denote the transmitted P and S wave, respectively, X refers

to the position, t is the time, h is the transmitted angle for P

wave, which is equal to the incident angle, and / is the

transmitted angle for S wave, which can be obtained by

Snell’s law. Thus, the theoretical solutions for transmission

coefficient in x direction can be obtained.


cient T1xj j of transmitted waves measured in the x direction

for obliquely incident P wave as a function of the incident

angle h, where kn ¼ ks ¼ 20 GPa/m. It can be found that

T1xj j from UDEC modeling agrees with the analytical

solutions. The largest computational error, which is equal

to 6.8 %, occurs when the incident angle is 45�. Although

the magnitude of computational error is accepted in

numerical modeling, the reason why the largest computa-

tional error occurs when the incident angle is around 45�needs to be explained. One possible reason is the error in

generating quadrilateral meshes near the joint. It is most

difficult to generate exact quadrilateral meshes near

inclined joint when the joint lies in the angular bisector

between the horizontal (axis x) and vertical (axis y)

directions.

2.5 Wave Transmission Across Multiple Intersecting

Joint Sets

A series of exploration tests were performed in Alvdalen,

Sweden between 2000 and 2001. The weight of TNT

ranges from 10 to 10,000 kg. The experimental data mea-

sured after exploration of 500 kg TNT are adopted to

validate the UDEC modeling results.

The spatial configuration of the field test in vertical

direction is shown in Fig. 7. The 500-kg TNT was deto-

nated in an underground chamber surrounded by jointed

rock mass. The vertical particle velocities were measured

by gauges buried in vertical boreholes, which are located 8,

16 and 32 m far away from the explosion chamber roof,

respectively. In the test site, the geological data for the

intact rock material are Young’s modulus 93.4 GPa, Pos-

sion’s ratio 0.27, and density 2,620 kg/m3 (Hansson and

Forsen 1997). There exist three major joint sets in the test

site. The spatial (orientation and spacing) and mechanical

(normal and shear stiffness) properties of joint sets are

10 20 30 40 50 600.5

0.6

0.7

0.8

0.9lT

1xl

θ, degree


Fig. 6 T1Xj j versus h for obliquely incident P wave transmission


Fig. 7 Spatial configuration of

field test site in vertical

direction and UDEC model. The

right figure is the zoom-in of the

area near the explosion chamber

roof in UDEC


123

illustrated in Table 1, based on the geological data from

Bakhtar and Jenus (1994); Bergab (2000) and Berglund

(2001).

As shown in Fig. 7, the four boundaries in UDEC model

are all non-reflection viscous ones except that velocity

boundary condition is adopted at the explosion chamber

roof. The velocity applied to the explosion chamber roof,

i.e., the incident wave in UDEC, is the same as the particle

velocity measured by the gauge at the chamber roof in filed

test and shown in Fig. 8.

The measured PPVs at the three locations from UDEC

and field test are exhibited in Fig. 9. The UDEC results

agree with field measured data for engineering problems. It

should be noted that joint planes are three-dimensional in

nature, while UDEC is a two-dimensional code. And this

will result in difference between field test data and UDEC

modeling results.

3 Parametric Studies on Wave Transmission Across

Multiple Intersecting Joint Sets

When multiple joint sets exist, wave transmission across

jointed rock masses is so complicated that analytical

solutions are impossible to obtain and experiments are

difficult to conduct. Therefore, numerical modeling is

an appropriate approach to address these complicated

problems.

The UDEC model is illustrated in Fig. 10 for a given

jointed rock mass covering a 20 m wide and 20 m high

area . The incident plane wave is applied at the left

boundary of the jointed rock mass. In rock engineering,

plane wave exists in the far field from the point or line

dynamic source, or near field from the line dynamic source.

Viscous non-reflection boundaries (Lyser and Kuhlemeyer

1969) are placed at all four boundaries of the model to

avoid wave reflections from the artificial boundaries.

However, in UDEC, the non-reflection boundary is not

perfect. A small amount of wave energy can still be

reflected from the boundary. In order to further eliminate

the effects of reflected wave on wave transmission across

the jointed rock mass, intact rock is added into the model

surrounding the jointed rock mass, which can delay the

arriving time of the reflected waves from boundaries. The

origin of the X–Y coordinates is located at the left bottom

of the model. Transmitted waves are measured at 19 points

evenly distributed at x = 50 m and y varying between 11

and 29 m. The properties of the rock material and the

incident wave are the same as those in Sect. 2.1. In addi-

tion, it is assumed in each case that mechanical properties

of all of the joints are the same and that the joint orientation

properties are the same for each joint set, which is rea-

sonable (Jaeger et al. 2007).

The magnitude of transmission coefficient across the

jointed rock mass Tj j is defined as the maximum super-

posed wave amplitude measured at the 19 points to that of

the incident wave. At each measuring point, the measured

transmitted wave is the result of the superposition of

transmitted P and S waves.

In order to describe the mechanical and spatial proper-

ties of the jointed rock mass, seven parameters are con-

sidered. They are the joint normal stiffness kn, the joint

shear stiffness ks, the nondimensional joint spacing n (the

ratio of the joint spacing to the S wave wavelength), the

joint spacing ratio r (the ratio of joint spacing of different

Table 1 Spatial and mechanical properties of rock joints

Dip angle

(�)aSpacing

(m)

Normal stiffness

(GPa)

Shear stiffness

(GPa)

143 0.6 128.62 50.67

0 1.2 64.31 25.34

5 0.6 128.62 50.67

a The dip angle used in UDEC is defined as the angle of the joint

plane relative to the positive x direction in counterclockwise

0.000 0.002 0.004 0.006 0.008 0.010 0.012-1.0

-0.5

0.0

0.5

1.0

1.5

Par

ticle

vel

osity

, m/s

Time, s

Fig. 8 Measured particle velocity history applied to explosion

chamber roof. It is adopted as the incident wave in UDEC modeling

5 10 15 20 25 30 350.00

0.05

0.10

0.15

0.20

0.25

PP

V, m

/s

Distance from explosion chamber roof, m

Field test data

UDEC modeling results

Fig. 9 PPVs from field test and UDEC modeling. They are measured

at locations 8, 16, and 32 m away from the explosion chamber roof


123

joint sets), the intersecting angle of different joint sets a(the angle smaller than 90�, as shown in Fig. 10), the

incident angle b (the angle between the incident wave and

the bisector of the intersecting angle, as shown in Fig. 10),

and the number of joint sets M. It should be noted that the

actual joint mechanical properties and spatial properties are

more complex, although there have been already seven

parameters here.

3.1 Parametric Studies on kn

In order to study the effects of kn on Tj j, other parameters

are fixed: ks ¼ 30 GPa/m, n ¼ 1 (2.25 m), r = 1, a ¼ 90�,b ¼ 0�, M = 2. Figure 11 shows the corresponding UDEC

model.

Figure 12 shows Tj j versus kn. It can be found that Tj jincreases monotonously with increasing kn, which is

understandable and the same as wave transmission across a

single joint and a joint set.

3.2 Parametric Studies on ks

In order to study the effects of ks on Tj j, the other

parameters are fixed: kn ¼ 30 GPa/m, n ¼ 1, r = 1,

a ¼ 90�, b ¼ 0�, M = 2. The UDEC model is the same as

that shown in Fig. 11.

Figure 13 shows Tj j versus ks. It can be found that Tj jincreases monotonously with increasing ks, which is the

same as Tj j versus kn. However, the influence of ks on Tj j isnot as great as that of kn. With increasing ks, the increment

of Tj j is relatively small compared with increasing kn. This

is because for obliquely incident P wave across each joint,

most transmitted energy is stored in the transmitted P wave

(Zhao et al. 2012). In addition, the effects of kn on the

transmission coefficient across a single joint is more sig-

nificant than those of ks. Figure 14 illustrates the effects of

kn and ks on the amplitude of transmitted P wave for P

wave propagation across a single joint with both small and

large incident angles (h = 10� and 80�), where the incident

wave and rock properties are the same as those used in this

section. It can be found that for P wave incidence, the

dependence of transmission coefficients on kn is much

greater than that on ks.

3.3 Parametric Studies on n

In order to study the effects of n on Tj j, the other param-

eters are fixed: kn ¼ ks ¼ 30 GPa/m, r = 1, a ¼ 90�,b ¼ 0�, M = 2. The UDEC model is shown in Fig. 11 for

the case of n ¼ 1.

Fig. 10 The scheme of UDEC

model for wave transmission

across a given jointed rock

mass, where a is the intersecting

angle, b is the incident angle

Fig. 11 UDEC model (n ¼ 1, r = 1, a ¼ 90�, b ¼ 0�, M = 2)


123

Figure 15 shows Tj j versus n. It can be found that, with

increasing n, Tj j first increases rapidly to the maximum

value, then it decreases, and finally it increases slowly. The

change of Tj j with n is due to two reasons. One is the

multiple wave reflections among joints, and the other is the

change of the number of joints included in the rock mass of

fixed size. The effects of multiple wave reflections among

joints on the transmission coefficients are shown in Fig. 4

for wave transmission across a joint set. Their effects on

Tj j should be similar. Hence, Tj j first increases rapidly to

the maximum value, and then it decreases to the minimum

value with increasing n. When n is large, multiple wave

reflections among joint have little effect on Tj j, and the

increase of Tj j with increasing n is due to the decreasing

number of joints included in the rock mass.

3.4 Parametric Studies on r

In order to study the effects of r on Tj j, the other param-

eters are fixed: kn ¼ ks ¼ 30 GPa/m, the nondimensional

joint spacing of one of the two joint sets n ¼ 1, a ¼ 90�,

b ¼ 0�, M = 2. The UDEC model is shown in Fig. 11 for

the case of r ¼ 1.

Figure 16 shows Tj j versus r. It can be found that Tj jfirst decreases to the minimum value before it increases.

Similar with the effects of n on Tj j, the change of Tj j with n

0 50 100 150 2000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8lT

l

kn, GPa/m

Fig. 12 Tj j versus kn, where ks ¼ 30 GPa/m, n ¼ 1, r = 1, a ¼ 90�,b ¼ 0�, M = 2

0 50 100 150 2000.15

0.20

0.25

0.30

0.35

0.40

lTl

ks, GPa/m

Fig. 13 Tj j versus ks, where kn ¼ 30 GPa/m, n ¼ 1, r = 1, a ¼ 90�,b ¼ 0�, M = 2. b

0.0 50.0G 100.0G 150.0G 200.0G0.0

0.2

0.4

0.6

0.8

1.0

Mag

nitu

de fo

r th

e tr

ansm

itted

P w

ave

kn, Pa

θ=10o

θ=80o

(a)

(b)

0.0 50.0G 100.0G 150.0G 200.0G0.0

0.2

0.4

0.6

0.8

1.0

The

am

plitu

de o

f tra

nsm

itted

P w

ave

ks, Pa/m

θ=10o

θ=80o

Fig. 14 The amplitude of transmitted P wave for obliquely incident P

wave across a single joint with different incident angles (h ¼ 10� and

80�) versus: a kn, where ks = 30 GPa/m; b ks, where kn = 30 GPa/m

0.0 0.5 1.0 1.5 2.00.2

0.3

0.4

0.5

0.6

0.7

lTl

ξ

Fig. 15 Tj j versus n, where kn ¼ ks ¼ 30 GPa/m, r = 1, a ¼ 90�,b ¼ 0�, M = 2


123

may be caused by two factors. One is the multiple wave

reflections among joints, and the other is the number of

joints in the rock mass. When r is small, the effects of

multiple wave reflections on Tj j decrease with increasing r,

and hence Tj j decreases. When r is large, multiple wave

reflections have little effect on Tj j; and with increasing r,

the joint number of one of the two joint sets decreases and

hence Tj j increases.

As seen from Figs. 15, 16, the effects of r on Tj j are not

as great as n. It is because different from n, with changing

r, only the joint spacing a joint set varies, while the joint

spacing of the other set is fixed.

3.5 Parametric Studies on a

In order to study the effects of a on Tj j, the other param-

eters are fixed: kn ¼ ks ¼ 30 GPa/m, n ¼ 1, r = 1, b ¼ 0�,M = 2. The UDEC model is shown in Fig. 17 for the case

of a ¼ 40�.Figure 18 shows Tj j versus a. It can be found that with

increasing a, Tj j decreases monotonously. This is because

the incident angle upon each joint has great effects on the

wave transmission. Wave energy is transmitted more when

the incident angle is large. In order to explain this, Fig. 19

shows the UDEC modeling results of the magnitude of

transmission coefficients for the superposed transmitted

P and S waves across a single joint set as a function

of incident angles for P wave incidence, where

kn ¼ ks ¼ 30 GPa/m, n ¼ 1. It can be found that the

magnitude of transmission coefficients across a joint set

increases with increasing incident angles. This is coinci-

dent with the conclusion of in situ tests that waves prop-

agating along the path across joints have much smaller

amplitudes than those propagating along the direction

parallel to the joints (King et al. 1986; Hao et al. 2001).

3.6 Parametric Studies on b

In order to study the effects of b on Tj j, the other param-

eters are fixed: kn ¼ ks ¼ 30 GPa/m, n ¼ 1, r = 1,

a ¼ 90�, M = 2. The corresponding UDEC model is

shown in Fig. 20 for the case of b ¼ 45�.

0.0 0.5 1.0 1.5 2.0 2.5 3.00.25

0.30

0.35

0.40

0.45lT

l

r

Fig. 16 Tj j versus r, where kn ¼ ks ¼ 30 GPa/m, n ¼ 1, a ¼ 90�,b ¼ 0�, M = 2


0 20 40 60 800.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

lTl

α,

Fig. 18 Tj j versus a, where kn ¼ ks ¼ 30 GPa/m, n ¼ 1, r = 1,

b ¼ 0�, M = 2

0 20 40 60 800.3

0.4

0.5

0.6

0.7

The

mag

nitu

de o

f tra

nsm

issi

onco

effic

ient

acr

oss

a jo

int s

et

Incident angle, degree

Fig. 19 The magnitude of transmission coefficients across a joint set

for P wave incidence as a function of incident angle, where

kn ¼ ks ¼ 30 GPa/m, n ¼ 1


123

Figure 21 shows Tj j versus b. It can be found that with

increasing b, Tj j first decreases to the minimum value at

b ¼ 45� before it increases. The curve in Fig. 21 is sym-

metrical with respect to the axis b ¼ 45�. This is because

the joint spatial configuration in the UDEC model for bvarying from 0� to 90� is symmetrical with respect to the

diagonal of the rock mass square, which corresponds to the

bisector of the intersecting angle when b ¼ 45�. For

example, the joint spacing configuration at b ¼ 20� and

that at b ¼ 70� are symmetrical with the diagonal of the

rock mass square.

That Tj j reaches the minimum value at b ¼ 45� but the

maximum values at b ¼ 0� and b ¼ 90� is because the

incident angle upon each joint determines the wave trans-

mission. The wave attenuates most when the joint is per-

pendicular to the incident wave direction, as shown in

Fig. 19. Although there is another joint set parallel to the

incident wave direction when b ¼ 45�, the joint set per-

pendicular to the incident wave directions dominates Tj j.

3.7 Parametric Studies on M

In order to study the effects of M on Tj j, some parameters

are fixed: kn ¼ ks ¼ 30 GPa/m, n ¼ 1, r = 1. The UDEC

model is shown in Fig. 22 for different cases of rock

masses including one joint set, two joint sets, three joint

sets with one set horizontal or vertical, and four joint sets.

Figure 23 shows Tj j versus M. It can be found that with

increasing M, Tj j decreases except when adding a hori-

zontal joint set. Similarly, it is because wave attenuation is

much less when it propagates along the direction parallel to

the joints than that propagating along the direction inter-

secting the joints.

4 Discussion

The transmitted waves obtained at the measuring points in

Fig. 6 are the superposition of transmitted P and S waves,

or the vector sum of the waves in the X and Y directions.

The failure of engineering rock masses is usually domi-

nated by the peak particle velocity (PPV), which is the

amplitude of the superposed wave. Therefore, the super-

posed transmitted wave is more useful in the application to

rock engineering. However, it should be noted that the

oscillating direction of the superposed transmitted waves at

different times may be different.

In the previous studies, especially the theoretical studies,

on wave transmission across rock joints, the area of the

rock mass is not fixed, and only the change of parameters

like joint spacing or joint number are considered, where

Fig. 4 can be taken as an example. When n changes, the

other joint spacing parameters, e.g., number of joints, does

not change. To be contrary, the jointed rock mass used for

studying wave transmission across multiple joint sets, as

shown in Fig. 10, covers a fixed area. When spatial

parameters like n or r change, the number of joints also

changes correspondingly. It is because we would like to

study the seismic and dynamic responses of a certain given

scale of engineering rock masses.

From previous parametric studies, it is found that

parameters including kn, ks, n, and r have great effects on

Tj j. a and b also influence Tj j when M = 2, as shown in

Figs. 18, 21. However, it could be hypothesized that the

influence of a and b on Tj j becomes smaller with

increasing M, as the rock mass is extremely discretized by

joints.

In order to prove this hypothesis, Fig. 24 shows Tj jversus b, where kn ¼ ks ¼ 30 GPa/m, n ¼ 1, r = 1,

M = 4, and the joint sets is equally intersected with

intersecting angles 45�. However, the orientation of each

joint of each joint set is not the same. Randomness of the

orientation of the joints has also been considered. Standard


0 20 40 60 800.28

0.29

0.30

0.31

0.32

0.33

0.34

lTl

β, degree

Fig. 21 Tj j versus b, where kn ¼ ks ¼ 30 GPa/m, n ¼ 1, r = 1,

a ¼ 90�, M = 2


123

deviation from the mean of the joint inclination angle with

uniform probability distribution is set and equal to 10�.

Figure 25 shows Tj j versus b, where kn ¼ ks ¼ 30 GPa/

m, n ¼ 1, r = 1, M = 6, and the joint sets is equally

intersected with intersecting angles 30�. Similarly, standard

deviation from the mean of the joint inclination angle with

uniform probability distribution is set and equal to 10�.

It should be noted that b varies between 0 and 22.5� in

Fig. 24 and that b varies between 0 and 15� in Fig. 25,

because Tj j versus b is symmetrical or periodical for other

values of b, similar with the results in Sect. 3.6 and Fig. 21.

From Figs. 24, 25, we can find that the orientation of

joint, which is reflected by a and b, has little effect on Tj j,when the number of joint sets is large. Tj j mainly depends

on kn, n, ks and r, especially on the first two for P wave

incidence. When there are four joint sets (M = 4), the

largest variation percentage of Tj j with b is 21 %. For

M = 6, it is 8.4 %. Therefore, the effects of joint orienta-

tion on the transmission coefficients decrease with

increasing number of joint sets. It is claimed here that the

jointed rock mass can be treated to be isotropic in a two-

dimensional space, when the number of joint sets is equal

to or larger than 4.

In addition, it is found that the effects of n on Tj j are

much more significant than those of r. Therefore, it can be

Fig. 22 UDEC models for

jointed rock masses containing:

a one joints set with inclination

angle 45� (n ¼ 1); b two joint

sets (n ¼ 1, r = 1, a ¼ 90�,b ¼ 0�); c three joint sets by

adding one horizontal joint set

with n ¼ 1 to the model shown

in (b); d three joint sets by

adding one vertical joint set

with n ¼ 1 to the model shown

in (b); e four joint sets by

adding one horizontal joint set

and one vertical joint set with

n ¼ 1 to the model shown in (b)

1 2 3 4

0.20

0.25

0.30

0.35

0.40

0.45

0.50

lTl

M

Fig. 23 Tj j versus M, where kn ¼ ks ¼ 30 GPa/m, n ¼ 1, r = 1


123

claimed that the ratio of joint spacing to wavelength other

than the difference between the joint spacing of different

sets should be sufficiently taken into account in rock

engineering applications.

Natural joints are complex due to filling material, sur-

face roughness, and damage, etc. (Jaeger et al. 2007).

Nonlinear deformational behavior of rock joints exist,

especially under large-amplitude stress wave. Although

joint stiffness is widely adopted in rock mechanics and

rock engineering to describe its mechanical property, only

it is not accurate enough to fully describe the nonlinear

mechanical properties of rock joints. The objective of this

paper was to address the effect of multiple intersecting

joint sets (especially the intersecting joint spatial proper-

ties) on wave propagation. Seven parameters are consid-

ered to describe the mechanical and spatial properties of

the jointed rock mass. Although nonlinear properties of

rock joints can be modeled by UDEC, it is not considered

here for two reasons. First, there are already as many as

seven parameters to describe the mechanical and spatial

properties of the rock mass, and thus, it will be difficult

to conduct parametric study if we consider additional

nonlinear parameters. Second, wave propagation across

joints with nonlinear deformational behavior has been

separately studied, with respect to the Barton-Bandis

nonlinear joint model (Zhao et al. 2006b, 2008), surface

roughness (Nolte et al. 2000; Li et al. 2011), filling material

(Li et al. 2009; Zhu et al. 2011c, 2012), and damage

(Rossmanith et al. 1996).

5 Conclusions

Rock joints dominate wave transmission in rock masses.

Therefore, it is important to study wave propagation across

joints, which are often in the form of multiple intersecting

joint sets, in areas of underground explosion, underground

protection, and earthquake engineering.

The effects of mechanical and spatial properties of rock

joints, which include joint normal and shear stiffness,

incident angle, joint spacing and joint number, and inter-

secting joints on wave transmission in UDEC modeling are

verified through comparison with analytical solutions and

field measured data.

For a rock mass with multiple joint sets of a given size,

it is complex to describe its properties. In this paper, its

mechanical and spatial properties are described by seven

parameters including the joint normal stiffness kn, the joint

shear stiffness ks, the nondimensional joint spacing n, the

joint spacing ratio r, the intersecting angle of different joint

sets a, the incident angle b, and the number of joint sets M.

Through extensive parametrical studies on the seven

parameters, it is found that although all of them can

influence wave transmission, kn, n and a have greater

effects on transmission coefficient. Because for P wave, the

particle moving direction is mostly in normal direction, kn

has greater effects on wave transmission than ks does. That

wave transmission is more sensitive with n than r is due to

the fact that n determines the variation of all joint sets,

while r controls only the change of one set. Because waves

propagating along the path across joints attenuate more

than those propagating along the direction parallel to the

joints, wave transmission is more dependent on a than b.

The effects of M on wave transmission depend on the angle

between the added joint sets and the incident wave. In

addition, when M is large, the joint orientation, which is

described by a and b, has little effect on wave transmis-

sion; hence the jointed rock mass can be treated to be

isotropic in a two dimensional space.

It indicates that we should be more concerned with joint

spacing and joint intersecting angles in engineering eval-

uation. For example, in determining the site of under-

ground tunnels or underground oil and ammunition storage,

areas of larger joint spacing and intersecting angle are of

priority.

0 5 10 15 20 25

0.19

0.20

0.21

0.22

0.23lTl

β, degree

Fig. 24 Tj j versus b (M = 4), where kn ¼ ks ¼ 30 GPa/m, n ¼ 1,

r = 1

0 5 10 150.165

0.170

0.175

0.180

0.185

lTl

β, degree

Fig. 25 Tj j versus b (M = 6), where kn ¼ ks ¼ 30 GPa/m, n ¼ 1,

r = 1


123

In rock engineering, simple and straightforward deter-

mination of wave propagation across jointed rock mass is

crucial. The explosive shock wave propagation in rock

mass is mainly estimated by empirical equations (Dow-

ding 1985). The empirical estimation is accurate in the

order of magnitude. Large errors exist between the

empirical estimation and field measurement (Deng et al.

2012). Therefore, it is meaningful to develop new

empirical equations for engineering applications. Equiva-

lent representation of the jointed rock mass could be an

appropriate approach to meet both accuracy and simplic-

ity. In equivalent description of rock masses with multiple

intersecting joint sets, a great number of parameters could

be involved. Among them, joint intersecting angle, joint

spacing, and joint normal stiffness (explosive wave is

mainly P wave) must be included and given special

attention, while joint normal stiffness, joint spacing ratio,

and incident angle could be ignored. The anisotropy of the

rock mass could also be neglected when more joint sets

exist, e.g., more than four.

The joint spatial configuration in this paper is two-

dimensional. However, real joints are three-dimensional in

space. Besides, for joints filled with weak medium, e.g.,

saturated clay, in addition to joint stiffness, viscosity

should be taken into account, and joint thickness may be

important for high-frequency waves, e.g., ultrasonic waves

of frequency higher than 1 MHz. In addition, rock and joint

damage may occur for large-amplitude wave propagation

in the ambient area of the dynamic source, which will

dissipate wave energy and affect wave propagation.

Therefore, further numerical studies on wave propagation

across jointed rock masses as well as their validation by

experimental tests are necessary and recommended.

Acknowledgments We would like to acknowledge the two anon-

ymous reviewers and the Editor, Professor Giovanni Barla, for their

constructive comments. Dr. Yingxin Zhou is acknowledged for the

sharing of field measured data. This research is financially supported

by the Swiss National Science Foundation (200021-116536) and

National Natural Science Foundation of China (40702046).

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