a numerical study on wave transmission across multiple joints
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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: jbzhu@caltech.edu
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
Across a Single Joint
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.
Figure 3 shows the magnitude of transmission coeffi-
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
Analytical solutions UDEC results
Fig. 3 T1Sj j versus ks for normally incident S wave transmission
across a single joint
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.
Figure 4 shows the magnitude of transmission coeffi-
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
Across a Single Joint
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
1432 J. B. Zhu et al.
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.
Figure 6 shows the magnitude of transmission coeffi-
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
Analytical solutions UDEC results
Fig. 6 T1Xj j versus h for obliquely incident P wave transmission
across a single joint
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
A Numerical Study on Wave Transmission 1433
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
1434 J. B. Zhu et al.
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)
A Numerical Study on Wave Transmission 1435
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
1436 J. B. Zhu et al.
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
Fig. 17 UDEC model (n ¼ 1, r = 1, a ¼ 40�, 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
A Numerical Study on Wave Transmission 1437
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
Fig. 20 UDEC model (n ¼ 1, r = 1, a ¼ 90�, b ¼ 45�, M = 2)
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
1438 J. B. Zhu et al.
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
A Numerical Study on Wave Transmission 1439
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
1440 J. B. Zhu et al.
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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