An equivalent viscoelastic model for rock mass with parallel joints

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1 Click Here for Full Article JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115,, doi: /2008jb006241, 2010 An equivalent viscoelastic model for rock mass with parallel joints Jianchun Li, 1 Guowei Ma, 1 and Jian Zhao 2 Received 6 December 2008; revised 11 July 2009; accepted 27 October 2009; published 11 March [1] An equivalent viscoelastic medium model is proposed for rock mass with parallel joints. A concept of virtual wave source (VWS) is proposed to take into account the wave reflections between the joints. The equivalent model can be effectively applied to analyze longitudinal wave propagation through discontinuous media with parallel joints. Parameters in the equivalent viscoelastic model are derived analytically based on longitudinal wave propagation across a single rock joint. The proposed model is then verified by applying identical incident waves to the discontinuous and equivalent viscoelastic media at one end to compare the output waves at the other end. When the wavelength of the incident wave is sufficiently long compared to the joint spacing, the effect of the VWS on wave propagation in rock mass is prominent. The results from the equivalent viscoelastic medium model are very similar to those determined from the displacement discontinuity method. Frequency dependence and joint spacing effect on the equivalent viscoelastic model and the VWS method are discussed. Citation: Li, J., G. Ma, and J. Zhao (2010), An equivalent viscoelastic model for rock mass with parallel joints, J. Geophys. Res., 115,, doi: /2008jb Introduction [2] Rock mass usually consists of multiple, parallel planar joints, known as joint sets, which govern the mechanical behavior of the rock mass. The dynamic behavior and wave propagation across jointed rock mass are of great interest to geophysics, mining, and underground constructions. It is also significant to assess the stability and damage of rock structures under dynamic loads. Because of the discontinuity by the joints, the dynamic response of jointed rock mass is a complicated process. It is of significance to develop an efficient and explicit model to represent the dynamic property of the jointed rock mass. [3] Currently, the methods for analyzing the effect of joints on the properties of rock mass can be divided into two categories: (1) displacement discontinuity method (DDM) [Miller, 1977; Schoenberg, 1980] and (2) the effective moduli methods [White, 1983; Schoenberg and Muir, 1989; Pyrak-Nolte et al., 1990; Cook, 1992]. In the DDM the stresses across the interface are continuous, whereas the displacements across the interface are discontinuous. Generally, the DDM treats joints, particularly the dominant sets as discrete entities. It predicts well the effect of joints on the transmission of seismic waves [Pyrak-Nolte, 1988; Cook, 1992]. Successful applications of this method have been reported for the wave transmission across single joint [Miller, 1977; Pyrak-Nolte, 1988; Pyrak-Nolte et al., 1990; Cook, 1992; Zhao and Cai, 2001] and multiparallel 1 School of Civil and Environmental Engineering, Nanyang Technological University, Singapore. 2 Laboratory for Rock Mechanics, Ecole Polytechnique Federale de Lausanne, Lausanne, Switzerland. Copyright 2010 by the American Geophysical Union /10/2008JB joints [Cai and Zhao, 2000; Zhao et al., 2006a, 2006b]. In all these applications, the joints were considered to be linear or nonlinear elastic, and the rock between each two joints is intact and elastic. On the basis of the DDM, the derivation of wave propagation equations is straightforward and it is in a differential form [Cai and Zhao, 2000; Zhao et al., 2006a, 2006b], which may not display an explicit expression of the solutions. [4] The effective moduli methods predict the aggregate effects of many joints or systems of joints within a representative elementary volume, so as to make a continuum analysis of problems contractible. Using a static approach, Zhao et al. [2006a] deduced the effective normal joint stiffness in rock mass with parallel joints and small joint spacing. Pyrak-Nolte et al. [1990] derived the time delay between two joints using the DDM and obtained the effective velocity for a normal incident wave through a set of parallel joints. Cook [1992] showed that the effective moduli methods account for the effect of joints on wave velocities while ignoring their influence on wave dissipation. The demerits of the effective moduli methods [White, 1983; Schoenberg and Muir, 1989; Pyrak-Nolte et al., 1990; Zhao et al., 2006a] are that they simplify the discontinuous rock mass to an elastic medium, which is effective only if the frequency dependence and the discreteness of joints, or multiple reflections among the joints, are negligible. [5] Pyrak-Nolte et al. [1990] indicated the frequency dependence can be accounted for with an assumption of an equivalent viscoelastic medium. By conducting extensive laboratory tests on ultrasonic wave transmission across natural joints, Pyrak-Nolte [1988] and Pyrak-Nolte et al. [1990] suggested that the natural rock joints may possess elastic as well as viscous coupling across the interface. 1of10

2 2.1. Wave Equations for Linear Viscoelastic Medium [9] Figure 1 shows the equivalent mathematical model of the auxiliary spring placed in series with the Voiget model, which has the stress-strain relation as ðe a þ E v Þs þ h ve E ve a e ¼ 0; ð1þ Figure 1. Equivalent mechanical model of an auxiliary spring in series with Voiget model. The definition of a linear viscoelastic solid is that it is a material for which the stress and the strain components are related by linear differential equations which involve the stress, the strain, and their derivatives with respect to time [Kolsky, 1953]. The wave propagation in linear viscoelastic solids has been investigated by Kolsky [1953] and Tsai and Kolsky [1968], in which the Voiget solid model, the Maxwell solid model, and some more general solid models were proposed. However, how a viscoelastic model is to be applied for the jointed rock mass has yet to be explored. [6] This paper proposes an equivalent viscoelastic medium model for rock mass with parallel joints. It combines a linear viscoelastic solid model with the concept of virtual wave sources (VWSs), in which the frequency dependence and the discreteness of joints in rock mass are taken into account. The parameters in the equivalent model are derived by analysis of longitudinal wave (P-wave) propagation along one representative joint spacing using the displacement discontinuity and the equivalent medium methods. To verify the proposed equivalent medium model, the results of the transmitted waves through a set of equally spaced joints are compared with those from the DDM. The effects of the VWS are also demonstrated. 2. Equivalent Viscoelastic Medium Model for Rock Mass with Parallel Joints [7] Natural jointed rock mass is discontinuous and may include one or more joint sets. When an incident wave propagates across the rock mass, the transmitted wave depends on not only the properties of the incident wave, but also those of the joints such as the stiffness and spacing [Pyrak-Nolte et al., 1990; Cai and Zhao, 2000; Zhao et al., 2006a, 2006b]. [8] Besides the Voiget and the Maxwell solid models, two extended linear viscoelastic solid models are also used for solid medium to describe the stress-strain relation [Kolsky, 1953]. One is an auxiliary spring in parallel with the Maxwell model, and the other is an auxiliary spring placed in series with the Voiget model. By comparing the Maxwell model, Voiget model, and their extended forms, it is found that the auxiliary spring placed in series with the Voiget model is a more appropriate equivalent model for a rock mass with one joint set, which can display both the attenuation and the frequency dependence of the transmitted wave. To consider the effect of the wave reflections between joints, the concept of VWS is introduced. where s is the stress, e is the strain, E a is the Young s modulus of the intact rock, E v is the stiffness contributed by the joints, and h v is the viscosity. Considering the longitudinal motion equation for 1-D problem, there @x ; where r is the density of the medium, v is the particle velocity, and t is time. [10] Differentiating equation (1) with respect to x and substituting with equation (2) 2 v rh 2 þ r ð E a þ E 2 Z 2 ve 2 E v ve a dt ¼ ð3þ Defining t = h v /E v as the time of retardation of the Voiget element, when a trial solution has the form of v ¼ A expðbx ð2þ Þexp½iðwt axþš; ð4þ where A is the amplitude of the incident velocity wave; w = 2pf and f is the frequency of the wave, it is found that equation (3) will be solved if 8 ( " a ¼ rw2 Ea 2 þ E2 c w2 t 2 1=2 # ) 1 2E c E a 1 þ w 2 t 2 þ E a þ E c w 2 t 2 2 >< 1 þ w 2 t 2 ( " b ¼ rw2 Ea 2 þ E2 c w2 t 2 1=2 # ) 1 2E c E a 1 þ w 2 t 2 E a þ E c w 2 t >: þ w 2 t 2 where 1 E c ¼ 1 E a þ 1 E v : Here a gives the phase shift per unit length, and the minus sign of b indicates the wave attenuation. It is shown in equations (4) and (5) that the wave propagation in a viscoelastic solid is frequency-dependent and its amplitude attenuates during the wave propagation process Virtual Wave Source [11] Assume a rock mass contains one joint set, i.e., equally spaced multiple parallel joints. When a longitudinal incident wave reaches a joint, a transmitted wave and a reflected wave are created and propagate in two opposite directions to the neighbor joints as two new incident waves. ; ð5þ ð6þ 2of10

3 Figure 2. Scheme of jointed rock mass and equivalent medium. Multiple transmitted and reflected waves are repeatedly created among the joints. Because of the discreteness of the joints, the newly created waves have different amplitudes and phase shifts to the incident waves. Across the joint set, the final transmitted wave is the superposition of two parts, one is from the direct transmission of the initial incident wave and the other part is from the multiple reflections among the joints. Although the frequency dependence and wave attenuation have been shown in equations (4) and (5), the effect of the discreteness of joints on wave propagation in the viscoelastic solid still cannot be reflected in the two equations. [12] To solve this problem, the concept of VWS is proposed in the equivalent viscoelastic medium model. The VWS exists at each joint surface and produces a new wave (in the opposite direction of the incident wave) at each time when an incident wave propagates across the VWS. The distance between two adjacent VWSs is equal to the joint spacing S. The equivalent length of the medium is defined as the product of joint number N and the joint spacing S, i.e., NS. Figure 2 shows a rock mass with three parallel joints and the corresponding equivalent medium with and without VWS, where the equivalent length is 3S. The concept of VWS can be interpreted as that a reflected wave is created from the VWS when either a positive wave or a negative wave arrives at the VWS. [13] Assume there is an incident P-wave v I ðt; 0Þ ¼ A expðiwtþ ð7þ from the left side a of the equivalent medium in Figure 2. According to equation (4), along the direction of the incident wave the particle velocity at point b is v e ðt; S Þ ¼ A expðbsþexp½iðwt asþš; ð8þ where the phase shift of v e (t, S) and v I (t,0)isas. According to the energy conservation of the simple harmonic waves [Cook, 1992], the amplitude of the reflected wave at the interface b is A{1 [exp(bs)] 2 } 1/2, if the interface b is a discontinuous boundary. From the Kramer-Kronig relation (a statement of causality), any changes in the amplitude of a wave must be accompanied by a change in phase. Since the phase shift between the reflected and transmitted waves is 3of10

4 3.1. Single Joint Case [15] If an incident P-wave at the boundary with the form of v I = Aexp(iwt) propagates in a rock mass with one joint, the transmitted wave after the joint was derived and written as [Pyrak-Nolte et al., 1990; Cook, 1992] or v T1 ¼ 2k=z A exp i wt xw=c iw þ 2k=z ½ ð ÞŠ ð10þ 2k=z v T1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A exp½iðwt xw=c þ qþš; ð11þ w 2 þ ð2k=zþ 2 Figure 3. Relation among parameters (a) E v and (b) h v, with frequency w. p/2 [Pyrak-Nolte et al., 1990; Cook, 1992], the reflected wave at b can be expressed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v 0 e SÞ 1 ½expðbSÞŠ 2 exp½iðwt þ as p=2þš; ð9þ where v 0 e(t, S) is regarded as the wave produced from the VWS at b. Then, v e (t, S) and the created wave v 0 e(t, S) propagate along two opposite directions as new incident waves to the adjacent interfaces c and a, where new waves are repeatedly created and propagate to their adjacent interfaces. The transmitted wave at the right side d of the equivalent medium is a wave superposition of v e (t, 3S) arriving at different times, which is the summation of multiple waves created from the three VWSs and the transmitted wave from the incident wave v I (t, 0) propagating across the viscoelastic medium. 3. Derivation of the Parameters [14] In this study, the joint is assumed to be planar, large in extent, and small in thickness compared to the wavelength, and the joint and the intact rock are linear elastic. For the equivalent medium model in equations (7) (9), E a is a known parameter equal to the Young s modulus of the intact rock, E v and h v need to be determined by comparing the transmitted wave through the equivalent medium with the existing solutions of discontinuous rock mass. where k is the joint normal stiffness; z is the wave impendence and z = rc; C is the P-wave velocity in the intact medium; x is the length of the rock mass along the wave propagation path; q = arctan[w/(2k/z)]. When x = S, the transmitted wave can be expressed as equation (11) for a rock mass with one joint or equation (8) for the corresponding equivalent viscoelastic medium. Thus, the amplitude and phase in equation (8) should, respectively, be equal to those in equation (11), i.e., b ¼ 1 S ln 6 2k=z 7 >< 4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 w 2 þ ð2k=zþ 2 a ¼ w C þ 1 ; S arctan w >: 2k=z ð12þ where a and b are shown in equation (5) Parameter Determination from Single Joint Analysis [16] Defining g 1 and g 2 as the functions of the two parameters, E v and h v, equation (12) can be rewritten as 2 3 g 1 ðe v ; h v Þ ¼ b 1 g S ln 6 2k=z 7 4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 w 2 þ ð2k=zþ 2 : ð13þ g 2 ðe v ; h v Þ ¼ a w C 1 S arctan w 2k=z The Newton-like methods [Kelley, 2003] are adopted to generate a sequence {(E v ) n (h v ) n } till the converged solutions are obtained, ( ) ðe v Þ nþ1 ðe ¼ v Þ n G ðh v Þ nþ1 ðh v Þ 1 g 2 ðe v Þ n ; ðh v Þ n ; ð14þ n g 2 ðe v Þ n ; ðh v where G is a matrix as follows: 1 ðe v Þ G ¼ ðh >: ðe v Þ ðh v Þ n Þ n Þ n 9 >= : ð15þ >; In the following calculations, it is assumed that the rock density r is 2650 kg/m 3 and the P-wave velocity C is 4of10

5 Table 1. Coefficients for Curves Fittings of E v ~ w, h v ~ w in Figures 3a and 3b S = 1/10l S = 1/2l S = l B 1 (GPa) B 2 (GPa) B 3 (GPa) B 4 (MPa s) B 5 (MPa s) B 6 (MPa s) F 1 (Hz) F 2 (Hz) F 3 (Hz) F 4 (Hz) m/s, the joint normal stiffness k is 3.5 GPa/m. The parameters E v and h v are predicted with different incident wave frequency w and joint spacing S. Figures 3a and 3b, respectively, show the relations between E v and w, h v and w, when S is 1/10l, 1/2l, and l. It can be seen that either E v or h v depends on the incident wave frequency w and the joint spacing S. For a given S, E v and h v decrease with increasing w. For a given w, E v and h v increase with increasing S. The relation between E v and S, h v and S can be, respectively, derived by the least square regression method as the two exponential forms, and E v ¼ B 1 expðw=f 1 ÞþB 2 expðw=f 2 ÞþB 3 ð16þ h v ¼ B 4 expðw=f 3 Þþ B 5 expðw=f 4 Þþ B 6 ; ð17þ where B i and F j (i = 1 6, j = 1 4) are the coefficients from the curve fitting, which are listed in Table Verifications 4.1. Wave Propagation through Parallel Joints [17] On the basis of the DDM, Cai and Zhao [2000] used a diamond-shape characteristic method and Zhao et al. [2006a] used a similar triangle-shape characteristic method [Ewing et al., 1957; Bedford and Drumheller, 1994] to develop the solutions of particle velocities before and after joints. In the discontinuous medium models, it is assumed that a finite number of interfaces are located at integral values of 1/(CDt), where Dt is the time interval. The conjunction points of right- and left-running characteristics in the x-t plane are shown in Figure 4. According to the characteristic line method [Zhao et al., 2006a], the two relations between the particle velocities and the normal stresses at points p 1, p 2, and p 4 are derived as zv x n ; t jþ1 þ s x n ; t jþ1 ¼ zv þ x n1 ; t j þ s þ x n1 ; t j ; where v (x n, t j+1 ) and v + (x n, t j+1 ) are the particle velocities at time t j+1 before and after the joint p 2 p 3 ; s (x n, t j+1 ) and s + (x n, t j+1 ) are the normal stresses at time t j+1 before and after the joint p 2 p 3. [18] If an interface is a joint, such as p 2 p 3, with linear deformational behavior, the DDM applied at point p 3 is expressed as s þ x n ; t j u x n ; t j ¼ s x n ; t j u þ x n ; t j ¼ s xn ; t j ð20þ s x n ; t j ¼ ; ð21þ k where u (x n, t j ) and u + (x n, t j ) are the displacements at time t j before and after the joint p 2 p 3 ; s (x n, t j ) and s + (x n, t j ) are the normal stresses at time t j before and after the joint p 2 p 3. [19] When the time interval Dt is very small, differentiation of equation (21) with respect to time t gives v x n ; t j v þ 1 x n ; t j ¼ x n ; t j ¼ k s x n ; t jþ1 s xn ; t j Dt : ð22þ Hence, the stress at time t j+1 on the joint p 2 p 3 can be expressed as s x n ; t jþ1 ¼ s xn ; t j þ kdt v x n ; t j v þ x n ; t j : ð23þ Considering equation (20), v (x n, t j+1 ) and v + (x n, t j+1 ) are deduced from equations (18) and (19) as v x n ; t jþ1 ¼ v þ 1 x n1 ; t j þ z s x n1; t j s xn ; t jþ1 ð24þ v þ x n ; t jþ1 ¼ v 1 x nþ1 ; t j þ z s x nþ1; t j þ s xn ; t jþ1 ; ð25þ where s(x n, t j+1 ) is calculated from equation (23). along right-running characteristic line p 1 p 2 ð18þ zv þ x n ; t jþ1 s þ x n ; t jþ1 ¼ zv x nþ1 ; t j s x nþ1 ; t j ; along left-running characteristic line p 2 p 4 ð19þ Figure 4. Characteristic lines in x-t plane for wave propagation through two joints. 5of10

6 [20] If an interface is not a joint, such as p 5 p 6, equations (18) (20) are still valid, but the displacements before and after the interface are continuous. Equation (21) is changed into u x nþ2 ; t j ¼ u þ x nþ2 ; t j ¼ uxnþ2 ; t j : ð26þ Considering equations (20) and (26) and replacing x n1, x n, and x n+1, respectively, with x n+1, x n+2, and x n+3 in equations (18) (20), the particle velocity and stress at point p 5 can be derived from the combination of equations (18) and (19), i.e., 1 vx nþ2 ; t jþ1 ¼ 2 vþ x nþ1 ; t j þ 1 2z s x nþ1; t j s x nþ2 ; t jþ1 þ v x nþ3 ; t j s xnþ3 ; t j ð27þ z ¼ 2 vþ x nþ1 ; t j v x nþ3 ; t j þ 1 2 s x ð28þ nþ1; t j þ s xnþ3 ; t j : [21] For a joint interface p 2 p 3, the stress and particle velocity at the point p 2 can be obtained from the stresses and particle velocities at the points of p 1, p 3, and p 4 using equations (23) (25), whereas the stress and particle velocity at the point p 5 for a nonjoint interface p 5 p 6 are expressed from the stresses and particle velocities at the points of p 4, p 6, and p 7 using equations (27) and (28). Then, the dynamic response for a rock mass with parallel joints can be calculated by using differential numerical method, if the input velocity of v(x n1, 0) and initial conditions of v + (x i, 0), v (x i, 0), and s(x i,0)(i = n, n +1,n +2,...) are known Periodical Function Expression for an Arbitrary Incident Wave [22] Using the Fourier and inverse Fourier transforms, any arbitrary incident wave can be expressed as the sum of periodical functions. Assume a half-cycle sinusoidal wave applied at the left side a in Figure 2 is the incident wave, i.e., v I ðt; 0Þ ¼ I sin ð w 0tÞ ; when 0 t p=w 0 0 others ð29þ where I is the amplitude of the incident wave and equal to 1 m/s; w 0 is the angular frequency of the incident wave. To obtain the periodical function expression of the half-cycle sinusoidal wave, the incident wave of equation (29) is first transformed in the frequency domain by the Fourier transform, Fv ½ I ðt; 0ÞŠ ¼ p 1 ffiffiffiffiffi 2p ¼ p 2 ffiffiffiffiffi 2p Z p=w0 0 w 2 0 I sinðw 0 tþe iwt dt w 0 I pw cos e i pw w2 : ð30þ The harmonic waveform with innumerable cycles of equation (29) is derived using the inverse Fourier transform of equation (30) in the time domain, v I ðt; 0Þ ¼ F 1 ffv ½ I ðt; 0ÞŠg ¼ 2 Z þ1 w 0 I pw cos cos wt pw dw: p w2 0 w 2 0 ð31þ If the frequency interval Dw is sufficiently small, equation (31) can be rewritten as v Ia ðt; 0Þ ¼ 2I p X þ1 j¼1 ( w 0 pw j w 2 0 cos cos w j t pw ) j Dw : w2 j ð32þ When the number of the frequency w j is sufficiently large, so that the main frequencies in the frequency domain of the incident wave are included, v Ia (t, 0) is approximately equal to v I (t, 0), i.e., v Ia (t, 0)ffi v I (t, 0) Result Comparison and Verification of the Equivalent Medium Model [23] To verify the proposed equivalent viscoelastic medium model, the transmitted wave calculated by the proposed model is compared with that determined by the DDM. Assuming the incident velocity wave at the boundary a in Figure 2 has the form of equation (29) and w 0 =2p100 Hz, the transmitted wave v Tdd through the discontinuous rock mass with joint spacing S is obtained from equations (23) to (25), (27), and (28). To analyze the wave propagation across the equivalent medium, the half-cycle sinusoidal wave given by equation (32) is chosen as the incident wave. Substituting cos w j t p ¼ exp iw j t p þ exp iw j t p 2 ð33þ into equation (32) and considering the wave propagation equations (5) and (7) (9) for the equivalent medium model, the transmitted wave at the interface b in Figure 2c by the incident wave is " v eb ¼ Xþ1 IDw 2 w 0 pw j p w 2 j¼1 0 cos w2 j expðbsþcos w j t pw # ð34þ j as and the reflected wave at interface b is ( v 0 eb ¼ Xþ1 IDw 2 w 0 pw j p w 2 j¼1 0 cos w2 j qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ½expðbSÞŠ 2 cos w j t pw j as p ) : 2 ð35þ 6of10

7 [24] The transmitted waves v Tdd based on the DDM and v Te based on the equivalent medium model are compared in Figures 5 and 6, for different joint number and joint spacing, respectively. In Figure 5, the joint spacing is one-tenth of the elastic wavelength l and l =2pC/w 0, while the joint number changes from one to four to reflect the medium length effect. The transmission coefficients, max(v Te )/max(v I ), are about 0.67 for the one-joint case, for the two-joint case, for the three-joint case, and 0.59 for the four-joint case. In other words, the transmitted wave attenuates quickly when the rock mass has one and two joints while the attenuation of the transmitted wave becomes slow with increasing number of joints. [25] Figure 6 shows the transmitted waves and their waveforms versus the joint spacing or the joint frequency when two joints are considered. The time delay of the transmitted wave over the two joints with S = l is about s, which is equal to that of w = w0 as given in in equation (4). The transmitted wave depends mainly on the original incident wave and the effect of the reflected Figure 5. Comparison of transmitted waves obtained from displacement discontinuity method (DDM) and equivalent medium method (EMM) with different joint number (S = 1/10l). The forward wave v eb and the backward wave v 0 eb then move toward the interfaces c and a, respectively, as new incident waves. The interfaces a, b, and c then perform as VWSs which generate a backward wave once an incident wave reaches. The waves superpose with each other in the medium and the final transmitted wave v Te at the interface d is obtained. Figure 6. Comparison of transmitted waves obtained from DDM and EMM with different joint spacing S. 7of10

8 Figure 7. Effect of virtual wave source (VWS) on transmitted waveform (S = 1/10l, joint number = 2). waves between the two joints appears minor for a large joint spacing, e.g., when S = l. [26] It is found from Figures 5 and 6 that the waveforms of v Te agree very well with those of v Tdd for all the cases studied. The comparisons between v Te and v Tdd verify the equivalent medium model proposed in this study, and the equivalent medium model can effectively describe the longitudinal wave propagation in a rock mass with one set of parallel joints. 5. Discussions 5.1. Effect of VWS on Transmitted Waveform [27] Define v Te,1 as the transmitted wave based on the equivalent viscoelastic medium without considering the effects of the VWSs; and define v Te,2 as the transmitted wave due to the reflections of the VWSs. From equation (34), the transmitted wave v Te,1 can be expressed as " v Te;1 ¼ Xþ1 IDw 2 w 0 pw j p w 2 j¼1 0 cos w2 j expðnbsþcos w j t pw # ð36þ j NaS ; where N is the number of the VWSs. When two VWSs are in the equivalent medium and S = 1/10l, the curves of v Te,1, v Te,2, and v Te, which is the superposition of v Te,1 and v Te,2, are plotted in Figure 7. v Te in Figure 7 is exactly the same as the transmitted wave shown in Figure 5b. v Te,1 in Figure 7 is the further attenuation with time shift of the transmitted wave shown in Figure 5a which is from an equivalent medium with only one VWS. It implies that v Te,1 is purely from the original incident wave. [28] When the VWS spacing S is l and the length of an equivalent medium is 2l or 4l, the transmitted waves across the equivalent medium with and without the VWS are derived as shown in Figure 8. For both medium lengths, the waveform of v Te,1 is very similar to the first part of v Te. Comparison of v Te in Figures 7 and 8 indicates that the VWS spacing influences the transmitted waveform, while the effect of the number of VWS on the transmitted waveform is minimal when the VWS spacing is larger than a specific value. The discussion on the specific value, beyond which the main transmitted waves are independent of VWS spacing, is given in the following section Effect of VWS on Transmission Coefficient [29] When an incident wave in a form as given in equation (29) with f = 100 Hz propagates across an equivalent medium with VWSs, the relation between the transmission coefficient and the VWS spacing S can be deduced as shown in Figure 9. It is found that the transmission coefficient decreases with the increase of the number of the VWS for a given spacing S. For the three cases, the transmission coefficient increases rapidly to a peak first in a small VWS spacing, drops down, and then becomes a constant with increasing VWS spacing. When the VWSs are not taken into account in the media, the transmission coefficients are also calculated from equations (5) and (34) to be 0.48 for two VWSs, 0.31 for four VWSs, and 0.25 for six VWSs, which are the constant values for the transmission coefficients in Figure 9. Therefore, the dependence of the transmission coefficient on VWS spacing is not obvious when the VWS spacing is large, e.g., S > 0.24l for the equivalent medium with two VWSs, S > 0.46l for the equivalent medium with four VWSs, and S > 0.57l for the equivalent medium with four VWSs Effective Velocities [30] The effective velocity C e for the incident wave v I in an equivalent medium is a function of the medium length to the time difference between the two peak velocities of incident and transmitted waves, i.e., C e ¼ NS t T t I ; ð37þ Figure 8. Transmitted waves by using DDM and EMM without VWS (f = 100 Hz, S = l). 8of10

9 Figure 9. Transmission coefficients across equivalent medium with VWS. where t I and t T are the time spots for the peak velocities of v I and v Te, respectively. [31] From this study, the effective velocity of the equivalent medium is affected by the ratio of the wavelength l over the spacing S. For example, if the incident wave in the form of equation (29) has the frequency f = 100 Hz or l = 58.3 m and the spacing S is 1/10l, the effective velocity C e can be calculated from Figure 5. On the basis of equation (37), C e is about 3320 m/s for the one-joint case, which approaches to the effective velocity C eff = 3322 m/s given by Pyrak-Nolte et al. [1987], " 2 # 1 þ zw0 C eff ¼ C 1 þ zw0 2k 2k 2þ C S z 2k ; ð38þ where the interaction between joints and multiple reflections are ignored. C e of the two-joint case is calculated about 2780 m/s from Figure 5b, about 2365 m/s for the three-joint case from Figure 5c, and about 2180 m/s for the four-joint case from Figure 5d. It is clear that the effective velocity is sensitive to the ratio of l/s. The reason is that the peak values of the transmitted wave are affected by the reflected waves created by the VWSs, as shown in Figure 7. It is the advantage of the present equivalent medium model to accurately account for the discreteness of the joints. [32] If the joint spacing S is larger, e.g., equals to l, as shown in Figure 6c, the effective velocity C e is calculated about 5420 m/s which matches with the effective velocity C eff = 5421 m/s given by Pyrak-Nolte et al. [1987]. Therefore, if the VWS spacing is sufficiently large, the influence of the multiple reflections among joints on the main transmitted wave is minimal, and the effective velocity C e is the same as that given by Pyrak-Nolte et al. [1987]. On the other hand, if the VWS is not considered in the equivalent medium model, the effective velocity derived from this study agrees very well with the previous results by Pyrak-Nolte et al. [1987]. 6. Conclusions [33] An equivalent viscoelastic medium model is proposed in this paper for determining the wave transmission through a rock mass containing equally spaced parallel joints. In the proposed equivalent medium model, the linear viscoelastic property of the medium is combined with the concept of VWSs. [34] Most existing analytical and numerical studies on wave propagation in jointed rock mass approximate the medium as a continuous elastic solid with the effective Young s modulus based on either a quasi-static deformation approach for a composite material or an effective seismic/ acoustic velocity approach. These traditional equivalent models, as limited by the elastic assumption, are not able to describe the effect of the discreteness of the joints. On the basis of this study, the equivalent viscoelastic medium model not only predicts accurately the transmission coefficients, but also derives analytically the transmitted waveforms. [35] The proposed equivalent viscoelastic medium model also has obvious advantage over the DDM. When the effect of the VWSs is not prominent, the proposed equivalent model is much more efficient in analyzing wave propagations. When the VWSs are considered, the proposed model is still able to give analytical solutions of the wave propagations while not losing the efficiency and accuracy. [36] Although this study involved only simplified cases with equally spaced parallel joints, it demonstrated that the current viscoelastic equivalent medium model is able to produce results in wave propagation analysis as accurate as those from the displacement discontinuity models. This study with the explicit form of solutions can serve as a benchmark for verification of relative numerical or analytical studies of stress wave propagation through rock mass. Further exploration is underway to extend the current model for more complicated joint forms and incident waves. References Bedford, A., and D. S. 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10 Zhao, J., X. B. Zhao, and J. G. Cai (2006a), A further study of P-wave attenuation across parallel fractures with linear deformational behaviour, Int. J. Rock Mech. Min. Sci., 43(5), Zhao, X. B., J. Zhao, and J. G. Cai (2006b), P-wave transmission across fractures with nonlinear deformational behaviour, Int. J. Numer. Anal. Methods Geomech., 30(11), J. Li and G. Ma, School of Civil and Environmental Engineering, Nanyang Technological University, Singapore. (cgwma@ntu.edu.sg) J. Zhao, Laboratory for Rock Mechanics, Ecole Polytechnique Federale de Lausanne, CH-1015 Lausanne, Switzerland. 10 of 10