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Stress wave interaction with a nonlinear and slippery rock joint
International Journal of Rock Mechanics & Mining Sciences 48 (2011) 493–500
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Technical Note
Stress wave interaction with a nonlinear and slippery rock joint Jianchun Li a,b, Guowei Ma b,c,n, Jian Zhao d a
State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, China School of Civil and Environmental Engineering, Nanyang Technological University, Singapore School of Civil and Resource Engineering, University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia d Ecole Polytechnique Federale de Lausanne (EPFL), Laboratory for Rock Mechanics (LMR), CH-1015 Lausanne, Switzerland b c
a r t i c l e in f o Article history: Received 22 December 2009 Received in revised form 18 June 2010 Accepted 27 November 2010 Available online 20 January 2011
1. Introduction The interaction of stress wave and rock joints is crucial in rock engineering to assess the stability and damage of underground structures under dynamic loads. Since stress waves generally, obliquely impinge to the rock joints, analysis of the oblique interaction is of special interests to mining engineers, seismologists and geoscientists. Wave propagation across an interface of two elastic media has been analyzed in detail by Kolsky [1] to calculate the reflection and transmission coefficients, by using a simple harmonic wave in the particle movement equation and Hooke’s law. Based on the wave motion theory and the principle of conservation of momentum, the mechanism of longitudinal- (P-) and shear- (S-) waves propagating across an interface between the two media has been described by Johnson [2]. The above two studies established the relation between the wave propagation velocities and the angles of the incident, reflected and transmitted waves for a free boundary and welded interface. However, the natural rock joints are generally non-welded, large in extent with void spaces and asperities of contact, and have relative displacements (opening, closure and slip) under normal and shear stresses [3–7]. The displacement–discontinuous method has been considered to be the most effective method [8,9] for analyzing wave propagation through linear [5,10–13] or nonlinear joints [13–17]. Using this method, a rock joint was modeled as a displacement– discontinuous boundary of two planar elastic half-spaces to derive close-form solutions for P- and S-waves obliquely impinge the joint surface [5,10,11,13]. Besides the free boundary and welded
n Corresponding author at: School of Civil and Resource Engineering, University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia. E-mail address: [email protected] (G. Ma).
1365-1609/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijrmms.2010.11.013
interface, Daehnke [13] analyzed the wave interactions with many kinds of rock joint, e.g., the joints have friction and shear strength or normal and shear stiffness. In the above studies, the incident wave was only a harmonic wave and the mathematical equation for the wave interaction with a nonlinear rock joint was not completely expressed. Hence, the Fourier and inverse Fourier transforms are needed to represent an incident wave with an arbitrary waveform. The present study aims to investigate an explicit law for any incident wave propagation with oblique angles across a complex rock joint with nonlinear and slippery behavior. Besides the effect from the incident waves, the interaction between the stress wave and rock joints is also affected by the mechanical properties of the joints. The normal and shear properties of joints can be described by linear elastic models [3,5–7] if the joint is dry and the magnitude of the stress wave is too small to mobilize its nonlinear deformation. However, the complete normal deformation behavior of a rock joint has been experimentally found to be generally nonlinear [4,18–21]. Among those models, Barton– Bandis model (B–B model) [19] has been widely used in representing joint normal property and analyzing wave propagation in jointed rock mass [12,15,22]. From the dynamic test results, a dynamic B–B model for the dynamic property of rock joints has been suggested [20]. In addition, the slip frictional behavior of rock joints also has certain influence on the wave propagation. A sliprate-dependent frictional model has been used in the analysis of S-wave attenuation across a single joint [8,23], by adopting an approximate analytical approach to simplify the nonlinearity into linearity of the joint shear behavior. The shear strength of a smooth joint was described by a normal effective stress and a frictional angle [3,6], and for most rock joints, the basic frictional angle varies in the range 25–351. Available shear strength models include the linear frictional model when the shear is on a horizontal smooth or an up-inclined smooth plane, the bi-linear shear strength model in which the cohesion coefficient is considered at high stress shearing
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off the asperities, the JRC–JCS empirical shear strength model in which the joint roughness coefficient, joint wall compressive strength and the drained residual friction angle are considered [3], and the JRC–JMC model as an extended JRC–JCS model by introducing the joint matching coefficient [6]. The present study employs the principle of conservation of momentum and the displacement discontinuous conditions along a rock joint to investigate the interaction of an obliquely incident P- or S-wave with a rock joint. The B–B model for the normal behavior and the coulomb-slip model for the shear behavior of the rock joint are adopted. The joint is assumed to be planar, dry, large in extent and small in thickness relative to the incident wavelength. The interaction is mathematically expressed as wave propagation equations for both incident P- and S-waves. Verification is conducted by comparing the analytical solutions with the existing results in literatures. Discussions are carried out with respect to the effects of the joint mechanical properties and incident waves on wave transmission and reflection. The proposed method can be straightforwardly extended to analyze the interaction of joints with other nonlinear and slippery models and incident waves with arbitrary waveforms.
2.2. Stresses and particle velocities along the joint (incident P-wave) When a thin beam of an incident P-wave impinges on the left side of a joint, a small element ABC composed by AB, AC and BC, which are the left side of the joint, the wave-front and side, respectively, as shown in Fig. 2a, can be considered. There are also some other small elements on the sides of the joint, which are formed by the transmitted and reflected P- and S-waves and the beam sides. In Fig. 2b–e, lines BD and BE are the wave-front of the reflected P- and S-waves, while lines BF and BG are the wave-front of the transmitted P- and S-waves, respectively. Since the present two-dimensional problem can be considered as a plane strain problem, the stress on the line BC of the side of the incident wave is [n/(1 n)]sIp, where sIp is the normal stress of the incident P-wave at its wave-front. Not considering the body force, the stresses on the element ABC, shown in Fig. 2a, should satisfy the equilibrium condition. So the stresses s1 and t1 on the left side of the joint for an element ABC can be derived. Similarly, the stresses si and ti (I¼2–5) on the two sides of the joint for the elements ABD, ABE, ABF and ABG in Fig. 2b–e can also be derived.
Reflected P-wave
Incident P-wave
Reflected S-wave
2. Problem formulation 2.1. Reflection and transmission for oblique incident waves on a rock joint When a P- or S-wave impinges a rock joint, both wave reflection and transmission take place, as shown in Fig. 1, where a and b are, respectively, the angles of the incident P- and S-waves. If the critical angles of the incident P- and S-waves are, respectively, defined to be ac and bc, there are 0 r a r ac and 0r b r bc, where ac ¼ 901 and bc ¼ sin1 ðcs =cp Þ from the Snell’s law, cp and cs are, respectively, the P- and S-waves propagation velocities in the intact medium. When the incident angles are greater than the critical angles, the emergence angles of reflected and transmitted waves are no longer real-valued, which will not be discussed in the paper. The two halfspaces of the rock media, beside the joint, are identical and considered to be an ideally elastic intact medium. In Fig. 1a, Ip stands for the incident P-wave, Rp and Rs represent reflected P- and S-waves and Tp and Ts are transmitted P- and S-waves, the symbols ‘‘ ’’ and ‘‘ + ’’ indicate the left and right sides of the joint. The incident P-wave propagates along the xz plane and the joint plane is on the xy plane. Similarly, in Fig. 1b, Is is the incident S-wave and the others are the same as those in Fig. 1a. Considering the Snell’s law, the reflection and transmission emergence angles must be equal to the incident angles for both the P- and S-waves incidences.
Fig. 2. Stresses on the wave-front and two sides of a joint (incident P-wave).
Incident P-wave
TP
RP RS α α
Incident S-wave x
x
β
α
β
RP TS
RS
β
TP α
α
β
Is -
+ Joint
TS
β
z
z IP
Transmitted S-wave
Transmitted P-wave
-
+ Joint
Fig. 1. Incident, reflected and transmitted waves on a rock joint.
J. Li et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 493–500
To simplify the problem, the compressive stress is defined to be positive in the present study. According to the conservation of momentum at the wave fronts, there are sIp ¼zpvIp, sRp ¼zpvRp, tRs ¼zsvRs, sTp ¼zpvTp and tTs ¼ zsvTs, where vIp, vRp and vTp are the particle velocities of the incident, reflected and transmitted P-waves, respectively; vRs and vTs are the particle velocities of the reflected and transmitted S-wave, respectively. According to Snell’s law, and defining zp ¼ rcp and zs ¼ rcs, where r is the density of the intact medium, the stresses at the left side of the joint can be expressed as
s ¼ zp cos 2b ðvIp þ vRp Þzs sin 2b vRs
ð1Þ
t ¼ zp sin 2b tan b=tan a ðvIp vRp Þzs cos 2b vRs
ð2Þ
495
2.3. Stresses and particle velocities along the joint (incident S-wave) Similarly, for an incident S-wave beam, shown in Fig. 3, the stresses s1 and t1 on the element ABC can also be derived. For this case, the stresses on the other small elements composed by the joint plane and corresponding waves are the same as those in Fig. 2b–e. According to the conservation of momentum at the wave fronts, there are tIs ¼ zsvIs, sRp ¼zpvRp, tRsv ¼zsvRsv, sTp ¼zpvTp and tTs ¼ zsvTs, where vIs is the particle velocity of the incident S-wave. Hence, the stresses at the left side of the joint can be expressed as
s ¼ zs sin 2b ðvIs vRs Þ þ zp cos 2b vRp
ð9Þ
t ¼ zs cos 2b ðvIs þvRs Þzp sin 2b tan b=tan a vRp
and the stresses at the right side of the joint are þ
ð3Þ
þ
ð4Þ
s ¼ zp cos 2b vTp þ zs sin 2b vTs t ¼ zp sin 2b tan b=tan a vTp zs cos 2b vTs
From Fig. 1a, the normal and tangential components of the velocities, v n and vt on the left side of the joint can, respectively, be expressed as,
ð10Þ
and the stresses at the right side of the interface are the same as given in Eqs. (3) and (4). The normal and tangential components of the velocities on the left side of the joint in Fig. 1b, respectively, satisfy v n ¼ sin b vIs cos a vRp þ sin b vRs
ð11Þ ð12Þ
v n ¼ cos a vIp cos a vRp þsin b vRs
ð5Þ
v t ¼ cos b vIs þ sin a vRp þ cos b vRs
v t ¼ sin a vIp þ sin a vRp þ cos b vRs
ð6Þ
and the normal and tangential components of the velocities on the right side of the joint in Fig. 1b have the same mathematical expression of Eqs. (7) and (8).
and the normal and tangential components of the velocities, vnþ and vtþ on the right side of the joint are vnþ ¼ cos a vTp þ sin b vTs
ð7Þ
þ
vt ¼ sin a vTp cos b vTs
2.4. Wave propagation equation across a single rock joint
ð8Þ Assume a rock joint has nonlinear and slippery behavior, e.g., the B–B model for the normal behavior and the Coulomb-slip model for the shear behavior, as shown in Fig. 4, where kni is the initial normal stiffness and dmax is the normal maximum allowable closure of the joint, ts is the shear strength, us is the critical relative slip, ks is the linear deformable slope and Dun and Dut are the closure and the relative slip of the rock joint. Fig. 4b shows that there are two portions consisted in the Coulomb-slip model: a linear elastic and a slip portions. The Coulomb-slip model assumes the same shear resistance in both directions which neglect the offset in the shear hysteretic loop [24,25]. If the shear stress on a joint-interface is smaller than ts, the joint has a linear elastic shear property with a slope of ks. When it reaches ts, the two joint sides slide over each other. Meanwhile, the stresses and displacements at the two sides of a rock joint satisfy the force equilibrium condition and the displacement–discontinuous boundary condition, i.e.
Joint Is B
C
Is
Is
1
β
1 A Is Joint Fig. 3. Stresses on the wave-front and left side of a joint (incident S-wave).
Joint
s ¼ s þ ¼ sd , t ¼ t þ ¼ t
τ
ks,
ð13Þ
σ
τs
τs ks 1 kni, dmax
O
us
Δuτ
1 O
kni dmax
Δun
Joint Fig. 4. Rock joint with nonlinear property: (a) joint model, (b) illustration of Coulomb-slip model, and (c) illustration of B–B model.
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J. Li et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 493–500
8 u unþ ¼ Dun ¼ s=ðkni þ s=dmax Þ > < n þ u t ut ¼ Dut ¼ t=ks , > : t ¼ 7 t ¼ 7 s tan f,
when9Dut 9 r us
ð14Þ
when9Dut 9 4 us
s
þ where u n and un are the normal displacements on the left and right þ sides of the joint; u t and ut are the shear displacements on the left and right sides; f is the frictional angle of the joint; the symbol ‘7’ represents both sliding direction of the joint-side; s is the normal stress on the joint-interface, which is composed of an initial compressive stress s0 and a dynamic normal stress sd caused by the incident wave, i.e., s ¼ s0 + sd, respectively. The shear strength ts consists of two portions: the initial shear strength ts0 and ts0 ¼ s0 tan f, the dynamic shear strength caused by the dynamic stress sd. Hence, ts can also be expressed as ts ¼ ts0 þ sd tan f. From Eqs. (3) and (14), s can be derived to be a function of ts0, vTp and vTs, i.e.
s ¼ ts0 =tan f þ zp cos 2b vTp þ zs sin 2b vTs
ð15Þ
When Eq. (14) is differentiated with respect to time t, there is 8 kni @s > v v þ ¼ ¼ ðk þ skni=d Þ2 si þD1tsi 2 > > max ni < nðiÞ nðiÞ ðkni þ s=dmax Þ @t þ 1 @t 1 ti þ 1 ti v v ¼ ¼ , when 9Dut 9 rus tðiÞ tðiÞ Dt ks @t ks > > > : ti ¼ 7 ts ¼ 7 ðts0 þ sd tan fÞ, when9Dut 94 us
ð16Þ
There are two possible joint shear deformation modes, i.e., the elastic mode and the relative slip mode when an incident P- or S-wave impinges a rock joint. Detail analysis can be categorized as four cases which are listed below. Case I: 9Dut9ous, Elastic mode, under an incident P-wave If the time interval Dt is very small, substituting Eqs. (1)–(4) into Eq. (13), and Eqs. (3)–(8) into Eq. (16) for 9Dut9 rus, the wave propagation equation in a matrix form can be derived and written as "
vRpðiÞ vRsðiÞ
#
" ¼ A1 BvIpðiÞ þ A1 C
vTpðiÞ
# ð17Þ
vTsðiÞ
and "
vTpði þ 1Þ vTsði þ 1Þ
#
" ¼ E1 FvIpðiÞ þ E1 G
vRpðiÞ
#
vRsðiÞ
" þ E1 H
vTpðiÞ
# ð18Þ
vTsðiÞ
where x ¼ ðkni þ s=dmax Þ2 =kni Dt, s is expressed as Eq. (15), and " A¼
" C¼ "
zp cos 2b
zs sin 2b
zp sin 2b tan b=tan a
zs cos 2b
zp cos 2b
zs sin 2b
zp sin 2b tan b=tan a
zs cos 2b
#
"
,
B¼
#
zp cos2b zp sin 2b tan b=tan a
# , #
zs sin 2b zp cos 2b , zp sin 2b tan b=tan a zs cos 2b " # " # x sin b x cos a x cos a , G¼ F¼ ks Dt sin a ks Dt cos b ks Dt sin a
E¼
" H¼
zp cos 2bx cos a
zs sin 2bx sin b
zp sin 2b tan b=tan aks Dt sin a
ks Dt cos bzs cos 2b
Eqs. (1)–(8) into Eqs. (13) and (16) for 9Dut94us, there is " # " # vRpðiÞ vTpðiÞ 1 1 ¼ A BvIpðiÞ þA C þ A1 D vRsðiÞ vTsðiÞ and " # vTpði þ 1Þ vTsði þ 1Þ
" 1
¼E
1
FvIpðiÞ þE
G
vRpðiÞ vRsðiÞ
#
" 1
þE
H
vTpðiÞ
ð20Þ
#
vTsðiÞ
ð21Þ
where A and B are the same as those in Eq. (19), while " # 0 zp cos 2b zs sin 2b , D¼ , C¼ 7 ts 0 0 F ¼ x cos a
E ¼ C, G¼
x cos a
x sin b
0 0 zp cos 2bx cos a H¼ 0
, zs sin 2bx sin b 0
ð22Þ
Hence, when a slip occurs, the particle velocities on the two sides of the rock joint can be calculated from Eqs. (20) and (21) with the parameter matrices given by Eq. (22). Case III: 9Dut9ous, Elastic mode, under an incident S-wave Similarly to Case I, putting Eqs. (3), (4), (9) and (10) into Eq. (13), the wave propagation equation for an incident S-wave across a single rock joint can also be expressed as the matrix form of Eqs. (17) and (18) with vIp(i) replaced by vIs(i), while h i h i ð23Þ Bu ¼ zs sin 2b zs cos 2b u, F ¼ x sin b ks Dt cos b u The other matrices, i.e., A, C–E, G and H, for the incident S-wave are the same as those in Eq. (19) and not to be repeatedly listed here. Case IV: 9Dut9Zus, Relative slip mode, under an incident S-wave Similarly to Case II, when a slip occurs, the wave propagation for an incident S-wave propagating across a complex joint is also derived. If Eqs. (3), (4), (9), (10) and Eqs. (7), (8), (11), (12) are put into Eqs. (13) and (16), the wave propagation equation is derived and can be expressed in the same matrix form of Eqs. (20) and (21) with vIp(i) replaced by vIs(i), while F ¼ x sin b, the other parameter matrices are the same as the aforementioned cases, i.e., A is the same as that in Eq. (19), B is the same as that in Eq. (23) and C–E, G and H are the same as those in Eq. (22). Hence, the wave propagation equations are derived as Eqs. (17)–(23) for incident P- and S-waves, where both the normal and shear nonlinear behaviors of rock joints are considered. When the boundary and initial conditions of the joint are known, the transmitted and reflected waves can then be analytically derived. The transmission coefficients, Tpc and Tsc, for P- and S-waves, and the reflection coefficients, Rpc and Rsc, for P- and S-waves, respectively, are defined as Tkc ¼
max9vTk 9 , max9vIk 9
Rkc ¼
max9vRk 9 , max9vIk 9
ðk ¼ p, sÞ,
ð24Þ
where k¼p, s stand for P- and S-waves respectively. # ð19Þ
Case II: 9Dut9 Zus, Relative slip mode, under an incident P-wave When the shear displacement of the joint-interface 9Dut9 is larger than us, the two sides of the joint occur a relative slip, and the normal and shear stresses on the two sides satisfy Eq. (14). Putting
3. Special cases 3.1. Normal incidence When an incident P-wave normally impinges on a rock joint, no slip occurs along the joint, i.e., 9Dut9¼0. Therefore, Eqs. (17) and
J. Li et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 493–500
3.3. Linearly deformable rock joint
(18) can be simplified to be z
p ¼ vTpðiÞ þ z2pDknit ðvIpðiÞ vTpðiÞ Þ kni þ dmax vTpðiÞ
:v
RpðiÞ þ vIpðiÞ
2 ð25Þ
¼ vTpðiÞ and vRsðiÞ ¼ vTsðiÞ
Similarly, for a normally impinging incident S-wave, Eqs. (17) and (18) can be simplified as 8 ¼ 2ks Dt=zs ðvIsðiÞ vTsðiÞ Þ þ vTsðiÞ , v > < Tsði þ 1Þ vRsðiÞ ¼ 8 ts0 =zs , when Dut Zus > : v þv ¼v , v ¼v ¼0 IsðiÞ
RsðiÞ
TsðiÞ
RpðiÞ
when Dut ous ð26Þ
If the amplitude of a stress wave is insufficient to mobilize the nonlinear and relative slip deformation of a joint, the linearly deformable model is valid for the joint [12]. For this case, the wave propagation equation with arbitrary impinging angles across a linearly deformable rock joint can be simplified, provided that the term s=dmax and the slippery of the joint are ignored in Eq. (14). Hence, the wave interaction with linear rock joints can also be obtained from the present study.
TpðiÞ
As can be seen that Eq. (25) is identical to the wave propagation equation given by Zhao and Cai [14], who used the characteristic line theory and obtained one-dimensional P-wave propagation law across a rock joint.
3.2. kni-N and ks-N A non-welded rock joint with an initial normal stiffness kni and a shear stiffness ks performs differently from that of a welded interface. However, they are equivalent when the non-welded rock joint has a very high normal and shear stiffness. The present analysis derives two transmitted and two reflected waves for a non-welded rock joint when an incident wave impinges the joint with an arbitrary angle. If kni-N and ks-N, from Eq. (16) there are vTp ¼vIp and vTs ¼vIs for incident P- and S-waves, respectively, which exactly coincides with that the incident waves propagate in an infinite space.
4. Discussions The interactions of an incident wave and a complex rock joint are further studied. Here, an incident P-wave VIP in a half-cycle sinusoidal form and an incident S-wave VIS in an one-cycle sinusoidal form are adopted, respectively, as the incident waves, i.e. ( VIP ¼ AI sinðotÞ, when t ¼ 0 p=o ð27Þ VIS ¼ AI sinðotÞ, when t ¼ 0 2p=o where AI is the amplitude of the incident waves; o ¼2pf and f is the frequency. 4.1. Transmitted and reflected waveforms In the following analysis, the parameters provided in [16] are adopted to be a set of reference parameters of the intact rock, i.e., the rock mass density is 2650 kg/m3, the P-wave velocity is 5830 m/s and the shear wave velocity is 2940 m/s. The frictional
kni=9.33 GPa/m
Reference parameters 0.3
Incident S-wave Transmitted P-wave Transmitted S-wave Reflected P-wave Reflected S-wave
Velocity (m/s)
0.2 0.1 0.0 -0.1
0.3
-0.2
0.1 0.0 -0.1 -0.2
-0.3 0.00
Incident S-wave Transmitted P-wave Transmitted S-wave Reflected P-wave Reflected S-wave
0.2 Velocity (m/s)
Tpði þ 1Þ
-0.3 0.01
0.02 0.03 Time (s)
0.04
0.05
0.00
0.01
Velocity (m/s)
0.04
0.05
0.3 Incident S-wave Transmitted P-wave Transmitted S-wave Reflected P-wave Reflected S-wave
0.2 0.1 0.0 -0.1
0.1 0.0 -0.1 -0.2
-0.3
-0.3 0.005
0.010 Time (s)
0.015
Incident S-wave Transmitted P-wave Transmitted S-wave Reflected P-wave Reflected S-wave
0.2
-0.2
0.000
0.02 0.03 Time (s) = 20°
f =200 Hz 0.3
Velocity (m/s)
8
497
0.00
0.01
0.02 0.03 Time (s)
Fig. 5. Transmitted and reflected waves of an incident S-wave for different parameters.
0.04
0.05
498
J. Li et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 493–500
angle of the joint is f ¼301, the initial normal stiffness kni ¼3.5 GPa/m and the shear stiffness ks ¼ kni zs =zp and the parameters for the incident waves have the amplitude AI ¼ 0.3 m/s, the frequency f¼50 Hz. A shear strength ratio is defined as d ¼ AI zs =ts0 and it is assumed to be d ¼3. When the joint satisfies the Coulomb-slip law, the transmitted and reflected waves of a normally incident S-wave with one-cycle sinusoidal form are calculated from the derived wave propagation equation in Section 2 and plotted in Fig. 5a. If the rock joint or the incident wave properties vary, the transmitted and reflected P- and S-waves are re-calculated and compared in Fig. 5b–d. Fig. 5a–c correspond to the normal incidence, while Fig. 5(d) shows a case with 201 impinging angle of the shear wave incidence. It can be observed from Fig. 5a to d that the transmitted and reflected waveforms are significantly influenced by the joint property and the incident wave parameters. For a normally incident S-wave, there only occur the transmitted and reflected S-waves, and the transmitted and reflected P-waves do not appear, which can be seen in Fig. 5a–c. However, when an incident S-wave obliquely impinges on the rock joint, four waves, i.e., the transmitted and reflected P- and S-waves are produced at the two sides of the joint, as shown in Fig. 5d. Comparing Fig. 5a and b, the relative slip occurs earlier for a rock joint with a higher initial stiffness than with a lower initial stiffness. Fig. 5a–c shows that for the cases of a normally incident S-wave, the peak value of the transmitted waves are not larger than AI =d or ts0 =zs and the positive and negative slip velocities are the same. While the peak value of the transmitted wave in Fig. 5d for the case of an obliquely incident S-wave is limited by ts =zs . The difference is primarily caused by the changes of the compressive normal stress along the joint, when the incident S-wave obliquely impinges on the joint.
Similarly, if the same parameters are adopted as for Fig. 5, but the incident wave changes to a P-wave, the transmitted and reflected P- and S-waves for different cases are re-calculated from the wave propagation equations in Section 2 and illustrated in Fig. 6a–d. It is observed that only two waves, i.e., transmitted and reflected P-waves are produced for the normally incidence cases, while four waves, i.e., two transmitted P- and S-waves and two reflected P- and S-waves occur for the obliquely incidence case, as shown in Fig. 6d. The calculated curves in Fig. 6 show that there is no relative slip occurring on the two sides of the joint for these cases. However, other parameters, such as the joint initial stiffness kni and the frequency f and the impinging angle a, still affect the stress wave and joint interaction. For example, the waveform of the transmitted P-wave in Fig. 6c expands wider at higher frequencies than those at lower frequencies. Comparing Fig. 6a and b, the larger initial stiffness kni, the more P-wave transmitted and the less P-wave reflected. By comparison, it is found that the transmitted and reflected waves shown in Fig. 5b are identical to those given by Zhao et al. [16], who used the one-dimensional characteristic line theory and the displacement discontinuity method to analyze S-wave normally impinging a rock joint with the Coulomb-slip property. Hence, it is once again verified the wave propagation equations derived in Section 2. 4.2. Transmission and reflection coefficients In this section, the effect of the joint behavior on wave transmission and reflection coefficients is studied, in terms of the initial normal stiffness kni and shear strength ratio d, where d is related to ts0. The parameters of the joint and the incident waves in Figs. 5d and 6d, for an incident S- or P-wave, are adopted.
kni=9.33 GPa/m
Reference parameters Incident wave Transmitted P-wave Transmitted S-wave Reflected P-wave Reflected S-wave
0.2
0.1
0.0
0.3
Velocity (m/s)
Velocity (m/s)
0.3
-0.1 0.00
Incident wave Transmitted P-wave Transmitted S-wave Reflected P-wave Reflected S-wave
0.2
0.1
0.0
-0.1 0.01
0.02 0.03 Time (s)
0.04
0.05
0.00
0.01
0.1 0.0
0.3
0.1 0.0
-0.1
-0.1
-0.2
-0.2
0.000 0.002 0.004 0.006 0.008 0.010 0.012 Time (s)
0.05
Incident wave Transmitted P-wave Transmitted S-wave Reflected P-wave Reflected S-wave
0.2 Velocity (m/s)
Velocity (m/s)
0.2
Incident wave Transmitted P-wave Transmitted S-wave Reflected P-wave Reflected S-wave
0.04
= 20°
f =200 Hz 0.3
0.02 0.03 Time (s)
0.00
0.01
0.02 0.03 Time (s)
Fig. 6. Transmitted and reflected waves of an incident P-wave for different cases.
0.04
0.05
J. Li et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 493–500
Incident S-wave
Incident S-wave
1.0
1.0 Tpc
0.8
Tsc
0.7
Rpc
0.6
Rsc
Transmission and reflection coefficients
Transmission and reflection coefficients
0.9
0.5 0.4 0.3 0.2 0.1 0.0 0.0
0.2
0.4
0.6
Tsc
0.8
Rpc
0.7
Rsc
0.6 0.5 0.4 0.3 0.2 0.1 0.0
1.0
0.8
Tpc
0.9
0
kni/(zpω)
2
Incident P-wave
10
Tpc Tsc
0.9
Tpc
0.8
Tsc
0.7
Rpc
Rpc
0.7
Rsc
0.6 0.5 0.4 0.3 0.2 0.1 0.4
8
1.0 Transmission and reflection coefficients
0.8
0.2
6
Incident P-wave
0.9
0.0 0.0
4
Shear strength ratio δ
1.0
Transmission and reflection coefficients
499
0.6
0.8
1.0
kni/(zpω)
Rsc
0.6 0.5 0.4 0.3 0.2 0.1 0.0 0
2
4 6 Shear strength ratio δ
8
10
Fig. 7. Effect of kni/(zpo) on the wave propagation coefficients. Fig. 8. Effect of shear strength ratio d on the wave propagation coefficients.
According to the wave propagation Eqs. (17)–(23), the transmission and reflection coefficients can be calculated from Eq. (24) with respect to different joint properties. Figs. 7a and 8a for an incident S-wave illustrate the relation of the transmission and reflection coefficients with different kni/(zpo) and d, respectively, and Figs. 7b and 8b are for an incident P-wave. As can be seen from Fig. 7, the wave transmission and reflection coefficients change with different values of kni/(zpo) for both incident P- and S-waves. The transmission coefficient Tsc by the incident S-wave and Tpc by the incident P-wave increase with an increasing kni/(zpo). Since ts0 ¼ AI zs =d, it can be concluded from Fig. 8 that the static shear strength ts0 influences only the transmission and reflection coefficients, Tkc and Rkc (k¼p, s) by an incident S-wave, while Tkc and Rkc (k¼p, s) by an incident P-wave keep constant for different d or ts0.
5. Conclusions An explicit wave propagation equation across a nonlinear and slippery rock joint is derived in the present study, where the principle
of the conservation of momentum at the two sides of an interface is adopted to analyze stress wave interaction with the rock joint. Both P- and S-waves obliquely impinging the joint are considered. The special cases show that the derived wave propagation equations can be simplified to those given in the literatures for normal incident waves. The comparison with the existing results demonstrates that the wave propagation equations derived in this paper are effective for the analysis of the body wave propagating across a complex rock joint with arbitrary impinging angles. It is also verified that the stress wave propagating across a rock joint is influenced not only by the normal and shear properties of the rock joint, but also by the properties of the incident wave, e.g., the frequency and the impinging angle of the incident P- and S-waves. When the incident wave impinges a complex joint, the relative slip of the joint is limited to the ratio of the shear strength of rock joints and the S-wave impedance, i.e., ts =zs . Compared to incident P-waves, incident S-waves are more likely to cause the relative slip of the joint. It should be mentioned that the wave propagation equations derived in the present study is valid only when the impinging angles are less than the critical angles. Further analysis is needed if the impinging angle exceeds this range.
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Acknowledgements The study is supported by Chinese National Science Research Fund (51025935, 11072257) and the Major State Basic Research Project of China (2010CB732001).
References [1] Kolsky H. Stress Waves in Solids. Oxford: Clarendon Press; 1953. [2] Johnson W. Impact Strength of Materials. London: Edward Arnold; 1972. [3] Barton N. Rock mechanics review, the shear strength of rock and rock joints. Int J Rock Mech Min Sci Geomech Abstr 1973;13:255–79. [4] Brown SR, Scholz CH. Closure of rock joints. J Geophys Res 1986;91:4939–48. [5] Cook NGW. Natural joint in rock: mechanical, hydraulic and seismic behavior and properties under normal stress. Int J Rock Mech Min Sci Geomech Abstr 1992;29(3):198–223. [6] Zhao J. Joint matching and shear strength, part A: joint matching coefficient (JMC). Int J Rock Mech Min Sci 1997;34:173–8. [7] Pyrak-Nolte LJ, Morris JP. Single fractures under normal stress: the relation between fracture specific stiffness and fluid flow. Int J Rock Mech Min Sci 2000;37:245–62. [8] Miller RK. An approximate method of analysis of the transmission of elastic waves through a frictional boundary. J Appl Mech -Trans ASME 1977;44(4): 652–6. [9] Schoenberg M. Elastic wave behavior across linear slip interfaces. J Acoust Soc Am 1980;68(5):1516–21. [10] Pyrak-Nolte LJ, Myer LR. Cook NGW. Anisotropy in seismic velocities and amplitudes from multiple parallel fractures. J Geophys Res 1990;95(B7): 11345–58. [11] Gu BL, Sua´rez-Rivera R, Nihei KT, Myer LR. Incidence of plane wave upon a fracture. J Geophys Res 1996;101(B11):25337–46.
[12] Zhao J, Zhao XB, Cai JG. A further study of P-wave attenuation across parallel fractures with linear deformational behavior. Int J Rock Mech Min Sci 2006;43: 776–88. [13] Daehnke A, Rossmanith HP. Reflection and refraction of plane stress waves at interfaces modelling various rock joints. Fragblast: Int J Blast Fragmentation 1997;1(2):111–231. [14] Zhao J, Cai JG. Transmission of elastic P-waves across single fracture with a nonlinear normal deformational behavior. Rock Mech Rock Eng 2001;34:3–22. [15] Zhao XB, Zhao J, Cai JG. P-wave transmission across fractures with nonlinear deformational behavior. Int J Numer Anal Methods Geomech 2006;30(11): 1097–112. [16] Zhao XB, Zhao J, Hefny AM, Cai JG. Normal transmission of S-wave across parallel fractures with coulomb slip behavior. J Eng Mech ASCE 2006;132(6): 641–50. [17] Li JC, Ma GW, Huang X. Analysis of wave propagation through filled rock joint. Rock Mech Rock Eng 2010;43(6):789–98. [18] Goodman RE. Methods of Geological Engineering in Discontinuous Rocks. 1st ed.. New York: West Pub; 1976. [19] Bandis SC, Lumsden AC, Barton NR. Fundamentals of rock fracture deformation. Int J Rock Mech Min Sci Geomech Abstr 1983;20(6):249–68. [20] Zhao J, Cai JG, Zhao XB, Li HB. Dynamic model of fracture normal behavior and application to prediction of stress wave attenuation across fractures. Rock Mech Rock Eng 2008;41(5):671–93. [21] Li JC, Ma GW. Experimental study of stress wave propagation across a filled rock joint. Int J Rock Mech Min Sci 2009;46(3):471–8. [22] Lonbard B, Piraux J. Modeling 1-D elastic P-waves in a fractured rock with hyperbolic jump conditions. J Comput Appl Math 2007;204:292–305. [23] Miller RK. The effects of boundary friction on the propagation of elastic waves. Bull Seismol Soc Am 1978;68(4):987–98. [24] Hsiung SM, Ghosh A, Ahola MP, Chowdhury AH. Assessment methodology for joint roughness coefficient determination. Int J Rock Mech Min Sci Geomech Abstr 1993;30(7):825–9. [25] Kana DD, Fox DJ, Hsiung SM. Interlock/friction model for dynamic shear response in natural jointed rock. Int J Rock Mech Min Sci Geomech Abstr 1996;33(4):371–86.
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Technical Note
Stress wave interaction with a nonlinear and slippery rock joint Jianchun Li a,b, Guowei Ma b,c,n, Jian Zhao d a
State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, China School of Civil and Environmental Engineering, Nanyang Technological University, Singapore School of Civil and Resource Engineering, University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia d Ecole Polytechnique Federale de Lausanne (EPFL), Laboratory for Rock Mechanics (LMR), CH-1015 Lausanne, Switzerland b c
a r t i c l e in f o Article history: Received 22 December 2009 Received in revised form 18 June 2010 Accepted 27 November 2010 Available online 20 January 2011
1. Introduction The interaction of stress wave and rock joints is crucial in rock engineering to assess the stability and damage of underground structures under dynamic loads. Since stress waves generally, obliquely impinge to the rock joints, analysis of the oblique interaction is of special interests to mining engineers, seismologists and geoscientists. Wave propagation across an interface of two elastic media has been analyzed in detail by Kolsky [1] to calculate the reflection and transmission coefficients, by using a simple harmonic wave in the particle movement equation and Hooke’s law. Based on the wave motion theory and the principle of conservation of momentum, the mechanism of longitudinal- (P-) and shear- (S-) waves propagating across an interface between the two media has been described by Johnson [2]. The above two studies established the relation between the wave propagation velocities and the angles of the incident, reflected and transmitted waves for a free boundary and welded interface. However, the natural rock joints are generally non-welded, large in extent with void spaces and asperities of contact, and have relative displacements (opening, closure and slip) under normal and shear stresses [3–7]. The displacement–discontinuous method has been considered to be the most effective method [8,9] for analyzing wave propagation through linear [5,10–13] or nonlinear joints [13–17]. Using this method, a rock joint was modeled as a displacement– discontinuous boundary of two planar elastic half-spaces to derive close-form solutions for P- and S-waves obliquely impinge the joint surface [5,10,11,13]. Besides the free boundary and welded
n Corresponding author at: School of Civil and Resource Engineering, University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia. E-mail address: [email protected] (G. Ma).
1365-1609/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijrmms.2010.11.013
interface, Daehnke [13] analyzed the wave interactions with many kinds of rock joint, e.g., the joints have friction and shear strength or normal and shear stiffness. In the above studies, the incident wave was only a harmonic wave and the mathematical equation for the wave interaction with a nonlinear rock joint was not completely expressed. Hence, the Fourier and inverse Fourier transforms are needed to represent an incident wave with an arbitrary waveform. The present study aims to investigate an explicit law for any incident wave propagation with oblique angles across a complex rock joint with nonlinear and slippery behavior. Besides the effect from the incident waves, the interaction between the stress wave and rock joints is also affected by the mechanical properties of the joints. The normal and shear properties of joints can be described by linear elastic models [3,5–7] if the joint is dry and the magnitude of the stress wave is too small to mobilize its nonlinear deformation. However, the complete normal deformation behavior of a rock joint has been experimentally found to be generally nonlinear [4,18–21]. Among those models, Barton– Bandis model (B–B model) [19] has been widely used in representing joint normal property and analyzing wave propagation in jointed rock mass [12,15,22]. From the dynamic test results, a dynamic B–B model for the dynamic property of rock joints has been suggested [20]. In addition, the slip frictional behavior of rock joints also has certain influence on the wave propagation. A sliprate-dependent frictional model has been used in the analysis of S-wave attenuation across a single joint [8,23], by adopting an approximate analytical approach to simplify the nonlinearity into linearity of the joint shear behavior. The shear strength of a smooth joint was described by a normal effective stress and a frictional angle [3,6], and for most rock joints, the basic frictional angle varies in the range 25–351. Available shear strength models include the linear frictional model when the shear is on a horizontal smooth or an up-inclined smooth plane, the bi-linear shear strength model in which the cohesion coefficient is considered at high stress shearing
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J. Li et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 493–500
off the asperities, the JRC–JCS empirical shear strength model in which the joint roughness coefficient, joint wall compressive strength and the drained residual friction angle are considered [3], and the JRC–JMC model as an extended JRC–JCS model by introducing the joint matching coefficient [6]. The present study employs the principle of conservation of momentum and the displacement discontinuous conditions along a rock joint to investigate the interaction of an obliquely incident P- or S-wave with a rock joint. The B–B model for the normal behavior and the coulomb-slip model for the shear behavior of the rock joint are adopted. The joint is assumed to be planar, dry, large in extent and small in thickness relative to the incident wavelength. The interaction is mathematically expressed as wave propagation equations for both incident P- and S-waves. Verification is conducted by comparing the analytical solutions with the existing results in literatures. Discussions are carried out with respect to the effects of the joint mechanical properties and incident waves on wave transmission and reflection. The proposed method can be straightforwardly extended to analyze the interaction of joints with other nonlinear and slippery models and incident waves with arbitrary waveforms.
2.2. Stresses and particle velocities along the joint (incident P-wave) When a thin beam of an incident P-wave impinges on the left side of a joint, a small element ABC composed by AB, AC and BC, which are the left side of the joint, the wave-front and side, respectively, as shown in Fig. 2a, can be considered. There are also some other small elements on the sides of the joint, which are formed by the transmitted and reflected P- and S-waves and the beam sides. In Fig. 2b–e, lines BD and BE are the wave-front of the reflected P- and S-waves, while lines BF and BG are the wave-front of the transmitted P- and S-waves, respectively. Since the present two-dimensional problem can be considered as a plane strain problem, the stress on the line BC of the side of the incident wave is [n/(1 n)]sIp, where sIp is the normal stress of the incident P-wave at its wave-front. Not considering the body force, the stresses on the element ABC, shown in Fig. 2a, should satisfy the equilibrium condition. So the stresses s1 and t1 on the left side of the joint for an element ABC can be derived. Similarly, the stresses si and ti (I¼2–5) on the two sides of the joint for the elements ABD, ABE, ABF and ABG in Fig. 2b–e can also be derived.
Reflected P-wave
Incident P-wave
Reflected S-wave
2. Problem formulation 2.1. Reflection and transmission for oblique incident waves on a rock joint When a P- or S-wave impinges a rock joint, both wave reflection and transmission take place, as shown in Fig. 1, where a and b are, respectively, the angles of the incident P- and S-waves. If the critical angles of the incident P- and S-waves are, respectively, defined to be ac and bc, there are 0 r a r ac and 0r b r bc, where ac ¼ 901 and bc ¼ sin1 ðcs =cp Þ from the Snell’s law, cp and cs are, respectively, the P- and S-waves propagation velocities in the intact medium. When the incident angles are greater than the critical angles, the emergence angles of reflected and transmitted waves are no longer real-valued, which will not be discussed in the paper. The two halfspaces of the rock media, beside the joint, are identical and considered to be an ideally elastic intact medium. In Fig. 1a, Ip stands for the incident P-wave, Rp and Rs represent reflected P- and S-waves and Tp and Ts are transmitted P- and S-waves, the symbols ‘‘ ’’ and ‘‘ + ’’ indicate the left and right sides of the joint. The incident P-wave propagates along the xz plane and the joint plane is on the xy plane. Similarly, in Fig. 1b, Is is the incident S-wave and the others are the same as those in Fig. 1a. Considering the Snell’s law, the reflection and transmission emergence angles must be equal to the incident angles for both the P- and S-waves incidences.
Fig. 2. Stresses on the wave-front and two sides of a joint (incident P-wave).
Incident P-wave
TP
RP RS α α
Incident S-wave x
x
β
α
β
RP TS
RS
β
TP α
α
β
Is -
+ Joint
TS
β
z
z IP
Transmitted S-wave
Transmitted P-wave
-
+ Joint
Fig. 1. Incident, reflected and transmitted waves on a rock joint.
J. Li et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 493–500
To simplify the problem, the compressive stress is defined to be positive in the present study. According to the conservation of momentum at the wave fronts, there are sIp ¼zpvIp, sRp ¼zpvRp, tRs ¼zsvRs, sTp ¼zpvTp and tTs ¼ zsvTs, where vIp, vRp and vTp are the particle velocities of the incident, reflected and transmitted P-waves, respectively; vRs and vTs are the particle velocities of the reflected and transmitted S-wave, respectively. According to Snell’s law, and defining zp ¼ rcp and zs ¼ rcs, where r is the density of the intact medium, the stresses at the left side of the joint can be expressed as
s ¼ zp cos 2b ðvIp þ vRp Þzs sin 2b vRs
ð1Þ
t ¼ zp sin 2b tan b=tan a ðvIp vRp Þzs cos 2b vRs
ð2Þ
495
2.3. Stresses and particle velocities along the joint (incident S-wave) Similarly, for an incident S-wave beam, shown in Fig. 3, the stresses s1 and t1 on the element ABC can also be derived. For this case, the stresses on the other small elements composed by the joint plane and corresponding waves are the same as those in Fig. 2b–e. According to the conservation of momentum at the wave fronts, there are tIs ¼ zsvIs, sRp ¼zpvRp, tRsv ¼zsvRsv, sTp ¼zpvTp and tTs ¼ zsvTs, where vIs is the particle velocity of the incident S-wave. Hence, the stresses at the left side of the joint can be expressed as
s ¼ zs sin 2b ðvIs vRs Þ þ zp cos 2b vRp
ð9Þ
t ¼ zs cos 2b ðvIs þvRs Þzp sin 2b tan b=tan a vRp
and the stresses at the right side of the joint are þ
ð3Þ
þ
ð4Þ
s ¼ zp cos 2b vTp þ zs sin 2b vTs t ¼ zp sin 2b tan b=tan a vTp zs cos 2b vTs
From Fig. 1a, the normal and tangential components of the velocities, v n and vt on the left side of the joint can, respectively, be expressed as,
ð10Þ
and the stresses at the right side of the interface are the same as given in Eqs. (3) and (4). The normal and tangential components of the velocities on the left side of the joint in Fig. 1b, respectively, satisfy v n ¼ sin b vIs cos a vRp þ sin b vRs
ð11Þ ð12Þ
v n ¼ cos a vIp cos a vRp þsin b vRs
ð5Þ
v t ¼ cos b vIs þ sin a vRp þ cos b vRs
v t ¼ sin a vIp þ sin a vRp þ cos b vRs
ð6Þ
and the normal and tangential components of the velocities on the right side of the joint in Fig. 1b have the same mathematical expression of Eqs. (7) and (8).
and the normal and tangential components of the velocities, vnþ and vtþ on the right side of the joint are vnþ ¼ cos a vTp þ sin b vTs
ð7Þ
þ
vt ¼ sin a vTp cos b vTs
2.4. Wave propagation equation across a single rock joint
ð8Þ Assume a rock joint has nonlinear and slippery behavior, e.g., the B–B model for the normal behavior and the Coulomb-slip model for the shear behavior, as shown in Fig. 4, where kni is the initial normal stiffness and dmax is the normal maximum allowable closure of the joint, ts is the shear strength, us is the critical relative slip, ks is the linear deformable slope and Dun and Dut are the closure and the relative slip of the rock joint. Fig. 4b shows that there are two portions consisted in the Coulomb-slip model: a linear elastic and a slip portions. The Coulomb-slip model assumes the same shear resistance in both directions which neglect the offset in the shear hysteretic loop [24,25]. If the shear stress on a joint-interface is smaller than ts, the joint has a linear elastic shear property with a slope of ks. When it reaches ts, the two joint sides slide over each other. Meanwhile, the stresses and displacements at the two sides of a rock joint satisfy the force equilibrium condition and the displacement–discontinuous boundary condition, i.e.
Joint Is B
C
Is
Is
1
β
1 A Is Joint Fig. 3. Stresses on the wave-front and left side of a joint (incident S-wave).
Joint
s ¼ s þ ¼ sd , t ¼ t þ ¼ t
τ
ks,
ð13Þ
σ
τs
τs ks 1 kni, dmax
O
us
Δuτ
1 O
kni dmax
Δun
Joint Fig. 4. Rock joint with nonlinear property: (a) joint model, (b) illustration of Coulomb-slip model, and (c) illustration of B–B model.
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J. Li et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 493–500
8 u unþ ¼ Dun ¼ s=ðkni þ s=dmax Þ > < n þ u t ut ¼ Dut ¼ t=ks , > : t ¼ 7 t ¼ 7 s tan f,
when9Dut 9 r us
ð14Þ
when9Dut 9 4 us
s
þ where u n and un are the normal displacements on the left and right þ sides of the joint; u t and ut are the shear displacements on the left and right sides; f is the frictional angle of the joint; the symbol ‘7’ represents both sliding direction of the joint-side; s is the normal stress on the joint-interface, which is composed of an initial compressive stress s0 and a dynamic normal stress sd caused by the incident wave, i.e., s ¼ s0 + sd, respectively. The shear strength ts consists of two portions: the initial shear strength ts0 and ts0 ¼ s0 tan f, the dynamic shear strength caused by the dynamic stress sd. Hence, ts can also be expressed as ts ¼ ts0 þ sd tan f. From Eqs. (3) and (14), s can be derived to be a function of ts0, vTp and vTs, i.e.
s ¼ ts0 =tan f þ zp cos 2b vTp þ zs sin 2b vTs
ð15Þ
When Eq. (14) is differentiated with respect to time t, there is 8 kni @s > v v þ ¼ ¼ ðk þ skni=d Þ2 si þD1tsi 2 > > max ni < nðiÞ nðiÞ ðkni þ s=dmax Þ @t þ 1 @t 1 ti þ 1 ti v v ¼ ¼ , when 9Dut 9 rus tðiÞ tðiÞ Dt ks @t ks > > > : ti ¼ 7 ts ¼ 7 ðts0 þ sd tan fÞ, when9Dut 94 us
ð16Þ
There are two possible joint shear deformation modes, i.e., the elastic mode and the relative slip mode when an incident P- or S-wave impinges a rock joint. Detail analysis can be categorized as four cases which are listed below. Case I: 9Dut9ous, Elastic mode, under an incident P-wave If the time interval Dt is very small, substituting Eqs. (1)–(4) into Eq. (13), and Eqs. (3)–(8) into Eq. (16) for 9Dut9 rus, the wave propagation equation in a matrix form can be derived and written as "
vRpðiÞ vRsðiÞ
#
" ¼ A1 BvIpðiÞ þ A1 C
vTpðiÞ
# ð17Þ
vTsðiÞ
and "
vTpði þ 1Þ vTsði þ 1Þ
#
" ¼ E1 FvIpðiÞ þ E1 G
vRpðiÞ
#
vRsðiÞ
" þ E1 H
vTpðiÞ
# ð18Þ
vTsðiÞ
where x ¼ ðkni þ s=dmax Þ2 =kni Dt, s is expressed as Eq. (15), and " A¼
" C¼ "
zp cos 2b
zs sin 2b
zp sin 2b tan b=tan a
zs cos 2b
zp cos 2b
zs sin 2b
zp sin 2b tan b=tan a
zs cos 2b
#
"
,
B¼
#
zp cos2b zp sin 2b tan b=tan a
# , #
zs sin 2b zp cos 2b , zp sin 2b tan b=tan a zs cos 2b " # " # x sin b x cos a x cos a , G¼ F¼ ks Dt sin a ks Dt cos b ks Dt sin a
E¼
" H¼
zp cos 2bx cos a
zs sin 2bx sin b
zp sin 2b tan b=tan aks Dt sin a
ks Dt cos bzs cos 2b
Eqs. (1)–(8) into Eqs. (13) and (16) for 9Dut94us, there is " # " # vRpðiÞ vTpðiÞ 1 1 ¼ A BvIpðiÞ þA C þ A1 D vRsðiÞ vTsðiÞ and " # vTpði þ 1Þ vTsði þ 1Þ
" 1
¼E
1
FvIpðiÞ þE
G
vRpðiÞ vRsðiÞ
#
" 1
þE
H
vTpðiÞ
ð20Þ
#
vTsðiÞ
ð21Þ
where A and B are the same as those in Eq. (19), while " # 0 zp cos 2b zs sin 2b , D¼ , C¼ 7 ts 0 0 F ¼ x cos a
E ¼ C, G¼
x cos a
x sin b
0 0 zp cos 2bx cos a H¼ 0
, zs sin 2bx sin b 0
ð22Þ
Hence, when a slip occurs, the particle velocities on the two sides of the rock joint can be calculated from Eqs. (20) and (21) with the parameter matrices given by Eq. (22). Case III: 9Dut9ous, Elastic mode, under an incident S-wave Similarly to Case I, putting Eqs. (3), (4), (9) and (10) into Eq. (13), the wave propagation equation for an incident S-wave across a single rock joint can also be expressed as the matrix form of Eqs. (17) and (18) with vIp(i) replaced by vIs(i), while h i h i ð23Þ Bu ¼ zs sin 2b zs cos 2b u, F ¼ x sin b ks Dt cos b u The other matrices, i.e., A, C–E, G and H, for the incident S-wave are the same as those in Eq. (19) and not to be repeatedly listed here. Case IV: 9Dut9Zus, Relative slip mode, under an incident S-wave Similarly to Case II, when a slip occurs, the wave propagation for an incident S-wave propagating across a complex joint is also derived. If Eqs. (3), (4), (9), (10) and Eqs. (7), (8), (11), (12) are put into Eqs. (13) and (16), the wave propagation equation is derived and can be expressed in the same matrix form of Eqs. (20) and (21) with vIp(i) replaced by vIs(i), while F ¼ x sin b, the other parameter matrices are the same as the aforementioned cases, i.e., A is the same as that in Eq. (19), B is the same as that in Eq. (23) and C–E, G and H are the same as those in Eq. (22). Hence, the wave propagation equations are derived as Eqs. (17)–(23) for incident P- and S-waves, where both the normal and shear nonlinear behaviors of rock joints are considered. When the boundary and initial conditions of the joint are known, the transmitted and reflected waves can then be analytically derived. The transmission coefficients, Tpc and Tsc, for P- and S-waves, and the reflection coefficients, Rpc and Rsc, for P- and S-waves, respectively, are defined as Tkc ¼
max9vTk 9 , max9vIk 9
Rkc ¼
max9vRk 9 , max9vIk 9
ðk ¼ p, sÞ,
ð24Þ
where k¼p, s stand for P- and S-waves respectively. # ð19Þ
Case II: 9Dut9 Zus, Relative slip mode, under an incident P-wave When the shear displacement of the joint-interface 9Dut9 is larger than us, the two sides of the joint occur a relative slip, and the normal and shear stresses on the two sides satisfy Eq. (14). Putting
3. Special cases 3.1. Normal incidence When an incident P-wave normally impinges on a rock joint, no slip occurs along the joint, i.e., 9Dut9¼0. Therefore, Eqs. (17) and
J. Li et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 493–500
3.3. Linearly deformable rock joint
(18) can be simplified to be z
p ¼ vTpðiÞ þ z2pDknit ðvIpðiÞ vTpðiÞ Þ kni þ dmax vTpðiÞ
:v
RpðiÞ þ vIpðiÞ
2 ð25Þ
¼ vTpðiÞ and vRsðiÞ ¼ vTsðiÞ
Similarly, for a normally impinging incident S-wave, Eqs. (17) and (18) can be simplified as 8 ¼ 2ks Dt=zs ðvIsðiÞ vTsðiÞ Þ þ vTsðiÞ , v > < Tsði þ 1Þ vRsðiÞ ¼ 8 ts0 =zs , when Dut Zus > : v þv ¼v , v ¼v ¼0 IsðiÞ
RsðiÞ
TsðiÞ
RpðiÞ
when Dut ous ð26Þ
If the amplitude of a stress wave is insufficient to mobilize the nonlinear and relative slip deformation of a joint, the linearly deformable model is valid for the joint [12]. For this case, the wave propagation equation with arbitrary impinging angles across a linearly deformable rock joint can be simplified, provided that the term s=dmax and the slippery of the joint are ignored in Eq. (14). Hence, the wave interaction with linear rock joints can also be obtained from the present study.
TpðiÞ
As can be seen that Eq. (25) is identical to the wave propagation equation given by Zhao and Cai [14], who used the characteristic line theory and obtained one-dimensional P-wave propagation law across a rock joint.
3.2. kni-N and ks-N A non-welded rock joint with an initial normal stiffness kni and a shear stiffness ks performs differently from that of a welded interface. However, they are equivalent when the non-welded rock joint has a very high normal and shear stiffness. The present analysis derives two transmitted and two reflected waves for a non-welded rock joint when an incident wave impinges the joint with an arbitrary angle. If kni-N and ks-N, from Eq. (16) there are vTp ¼vIp and vTs ¼vIs for incident P- and S-waves, respectively, which exactly coincides with that the incident waves propagate in an infinite space.
4. Discussions The interactions of an incident wave and a complex rock joint are further studied. Here, an incident P-wave VIP in a half-cycle sinusoidal form and an incident S-wave VIS in an one-cycle sinusoidal form are adopted, respectively, as the incident waves, i.e. ( VIP ¼ AI sinðotÞ, when t ¼ 0 p=o ð27Þ VIS ¼ AI sinðotÞ, when t ¼ 0 2p=o where AI is the amplitude of the incident waves; o ¼2pf and f is the frequency. 4.1. Transmitted and reflected waveforms In the following analysis, the parameters provided in [16] are adopted to be a set of reference parameters of the intact rock, i.e., the rock mass density is 2650 kg/m3, the P-wave velocity is 5830 m/s and the shear wave velocity is 2940 m/s. The frictional
kni=9.33 GPa/m
Reference parameters 0.3
Incident S-wave Transmitted P-wave Transmitted S-wave Reflected P-wave Reflected S-wave
Velocity (m/s)
0.2 0.1 0.0 -0.1
0.3
-0.2
0.1 0.0 -0.1 -0.2
-0.3 0.00
Incident S-wave Transmitted P-wave Transmitted S-wave Reflected P-wave Reflected S-wave
0.2 Velocity (m/s)
Tpði þ 1Þ
-0.3 0.01
0.02 0.03 Time (s)
0.04
0.05
0.00
0.01
Velocity (m/s)
0.04
0.05
0.3 Incident S-wave Transmitted P-wave Transmitted S-wave Reflected P-wave Reflected S-wave
0.2 0.1 0.0 -0.1
0.1 0.0 -0.1 -0.2
-0.3
-0.3 0.005
0.010 Time (s)
0.015
Incident S-wave Transmitted P-wave Transmitted S-wave Reflected P-wave Reflected S-wave
0.2
-0.2
0.000
0.02 0.03 Time (s) = 20°
f =200 Hz 0.3
Velocity (m/s)
8
497
0.00
0.01
0.02 0.03 Time (s)
Fig. 5. Transmitted and reflected waves of an incident S-wave for different parameters.
0.04
0.05
498
J. Li et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 493–500
angle of the joint is f ¼301, the initial normal stiffness kni ¼3.5 GPa/m and the shear stiffness ks ¼ kni zs =zp and the parameters for the incident waves have the amplitude AI ¼ 0.3 m/s, the frequency f¼50 Hz. A shear strength ratio is defined as d ¼ AI zs =ts0 and it is assumed to be d ¼3. When the joint satisfies the Coulomb-slip law, the transmitted and reflected waves of a normally incident S-wave with one-cycle sinusoidal form are calculated from the derived wave propagation equation in Section 2 and plotted in Fig. 5a. If the rock joint or the incident wave properties vary, the transmitted and reflected P- and S-waves are re-calculated and compared in Fig. 5b–d. Fig. 5a–c correspond to the normal incidence, while Fig. 5(d) shows a case with 201 impinging angle of the shear wave incidence. It can be observed from Fig. 5a to d that the transmitted and reflected waveforms are significantly influenced by the joint property and the incident wave parameters. For a normally incident S-wave, there only occur the transmitted and reflected S-waves, and the transmitted and reflected P-waves do not appear, which can be seen in Fig. 5a–c. However, when an incident S-wave obliquely impinges on the rock joint, four waves, i.e., the transmitted and reflected P- and S-waves are produced at the two sides of the joint, as shown in Fig. 5d. Comparing Fig. 5a and b, the relative slip occurs earlier for a rock joint with a higher initial stiffness than with a lower initial stiffness. Fig. 5a–c shows that for the cases of a normally incident S-wave, the peak value of the transmitted waves are not larger than AI =d or ts0 =zs and the positive and negative slip velocities are the same. While the peak value of the transmitted wave in Fig. 5d for the case of an obliquely incident S-wave is limited by ts =zs . The difference is primarily caused by the changes of the compressive normal stress along the joint, when the incident S-wave obliquely impinges on the joint.
Similarly, if the same parameters are adopted as for Fig. 5, but the incident wave changes to a P-wave, the transmitted and reflected P- and S-waves for different cases are re-calculated from the wave propagation equations in Section 2 and illustrated in Fig. 6a–d. It is observed that only two waves, i.e., transmitted and reflected P-waves are produced for the normally incidence cases, while four waves, i.e., two transmitted P- and S-waves and two reflected P- and S-waves occur for the obliquely incidence case, as shown in Fig. 6d. The calculated curves in Fig. 6 show that there is no relative slip occurring on the two sides of the joint for these cases. However, other parameters, such as the joint initial stiffness kni and the frequency f and the impinging angle a, still affect the stress wave and joint interaction. For example, the waveform of the transmitted P-wave in Fig. 6c expands wider at higher frequencies than those at lower frequencies. Comparing Fig. 6a and b, the larger initial stiffness kni, the more P-wave transmitted and the less P-wave reflected. By comparison, it is found that the transmitted and reflected waves shown in Fig. 5b are identical to those given by Zhao et al. [16], who used the one-dimensional characteristic line theory and the displacement discontinuity method to analyze S-wave normally impinging a rock joint with the Coulomb-slip property. Hence, it is once again verified the wave propagation equations derived in Section 2. 4.2. Transmission and reflection coefficients In this section, the effect of the joint behavior on wave transmission and reflection coefficients is studied, in terms of the initial normal stiffness kni and shear strength ratio d, where d is related to ts0. The parameters of the joint and the incident waves in Figs. 5d and 6d, for an incident S- or P-wave, are adopted.
kni=9.33 GPa/m
Reference parameters Incident wave Transmitted P-wave Transmitted S-wave Reflected P-wave Reflected S-wave
0.2
0.1
0.0
0.3
Velocity (m/s)
Velocity (m/s)
0.3
-0.1 0.00
Incident wave Transmitted P-wave Transmitted S-wave Reflected P-wave Reflected S-wave
0.2
0.1
0.0
-0.1 0.01
0.02 0.03 Time (s)
0.04
0.05
0.00
0.01
0.1 0.0
0.3
0.1 0.0
-0.1
-0.1
-0.2
-0.2
0.000 0.002 0.004 0.006 0.008 0.010 0.012 Time (s)
0.05
Incident wave Transmitted P-wave Transmitted S-wave Reflected P-wave Reflected S-wave
0.2 Velocity (m/s)
Velocity (m/s)
0.2
Incident wave Transmitted P-wave Transmitted S-wave Reflected P-wave Reflected S-wave
0.04
= 20°
f =200 Hz 0.3
0.02 0.03 Time (s)
0.00
0.01
0.02 0.03 Time (s)
Fig. 6. Transmitted and reflected waves of an incident P-wave for different cases.
0.04
0.05
J. Li et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 493–500
Incident S-wave
Incident S-wave
1.0
1.0 Tpc
0.8
Tsc
0.7
Rpc
0.6
Rsc
Transmission and reflection coefficients
Transmission and reflection coefficients
0.9
0.5 0.4 0.3 0.2 0.1 0.0 0.0
0.2
0.4
0.6
Tsc
0.8
Rpc
0.7
Rsc
0.6 0.5 0.4 0.3 0.2 0.1 0.0
1.0
0.8
Tpc
0.9
0
kni/(zpω)
2
Incident P-wave
10
Tpc Tsc
0.9
Tpc
0.8
Tsc
0.7
Rpc
Rpc
0.7
Rsc
0.6 0.5 0.4 0.3 0.2 0.1 0.4
8
1.0 Transmission and reflection coefficients
0.8
0.2
6
Incident P-wave
0.9
0.0 0.0
4
Shear strength ratio δ
1.0
Transmission and reflection coefficients
499
0.6
0.8
1.0
kni/(zpω)
Rsc
0.6 0.5 0.4 0.3 0.2 0.1 0.0 0
2
4 6 Shear strength ratio δ
8
10
Fig. 7. Effect of kni/(zpo) on the wave propagation coefficients. Fig. 8. Effect of shear strength ratio d on the wave propagation coefficients.
According to the wave propagation Eqs. (17)–(23), the transmission and reflection coefficients can be calculated from Eq. (24) with respect to different joint properties. Figs. 7a and 8a for an incident S-wave illustrate the relation of the transmission and reflection coefficients with different kni/(zpo) and d, respectively, and Figs. 7b and 8b are for an incident P-wave. As can be seen from Fig. 7, the wave transmission and reflection coefficients change with different values of kni/(zpo) for both incident P- and S-waves. The transmission coefficient Tsc by the incident S-wave and Tpc by the incident P-wave increase with an increasing kni/(zpo). Since ts0 ¼ AI zs =d, it can be concluded from Fig. 8 that the static shear strength ts0 influences only the transmission and reflection coefficients, Tkc and Rkc (k¼p, s) by an incident S-wave, while Tkc and Rkc (k¼p, s) by an incident P-wave keep constant for different d or ts0.
5. Conclusions An explicit wave propagation equation across a nonlinear and slippery rock joint is derived in the present study, where the principle
of the conservation of momentum at the two sides of an interface is adopted to analyze stress wave interaction with the rock joint. Both P- and S-waves obliquely impinging the joint are considered. The special cases show that the derived wave propagation equations can be simplified to those given in the literatures for normal incident waves. The comparison with the existing results demonstrates that the wave propagation equations derived in this paper are effective for the analysis of the body wave propagating across a complex rock joint with arbitrary impinging angles. It is also verified that the stress wave propagating across a rock joint is influenced not only by the normal and shear properties of the rock joint, but also by the properties of the incident wave, e.g., the frequency and the impinging angle of the incident P- and S-waves. When the incident wave impinges a complex joint, the relative slip of the joint is limited to the ratio of the shear strength of rock joints and the S-wave impedance, i.e., ts =zs . Compared to incident P-waves, incident S-waves are more likely to cause the relative slip of the joint. It should be mentioned that the wave propagation equations derived in the present study is valid only when the impinging angles are less than the critical angles. Further analysis is needed if the impinging angle exceeds this range.
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Acknowledgements The study is supported by Chinese National Science Research Fund (51025935, 11072257) and the Major State Basic Research Project of China (2010CB732001).
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