Transmission of Elastic P-waves across Single ... - Springer Link

Rock Mech. Rock Engng. (2001) 34 (1), 3±22

Rock Mechanics and Rock Engineering : Springer-Verlag 2001 Printed in Austria

Transmission of Elastic P-waves across Single Fractures with a Nonlinear Normal Deformational Behavior By

J. Zhao and J. G. Cai School of Civil and Structural Engineering, Nanyang Technological University, Singapore

Summary This paper presents a theoretical study on normally incident elastic P-wave transmission across single dry fractures with a nonlinear normal deformational behavior. The e¨ects of nonlinear fracture normal behavior on P-wave transmission are examined without the mixture of fracture shear behavior. The linear displacement discontinuity model for wave propagation across fractures is extended to a nonlinear model ± the hyperbolic elastic model (BB model). Numeric solutions of magnitudes of transmission …jTnon j† and re¯ection …jRnon j† coe½cients, for normally incident P-wave transmission across the nonlinear deformable fractures, are obtained and related to the closure behavior of fractures. Parametric studies are conducted to acquire an insight into the e¨ects of the nonlinear fracture normal deformation on P-wave transmission, in terms of initial normal sti¨ness and the ratio of current maximum closure to maximum allowable closure of the fractures, as well as the incident wave amplitude and frequency. Comparisons between the linear and nonlinear models are presented. It is shown that, jTlin j and jRlin j for the linear model are special solutions of jTnon j and jRnon j for the nonlinear model, when the incident wave amplitude is so low that the current maximum closure of fracture incurred during the wave transmission is much smaller, relative to the maximum allowable closure. In addition, the nonlinear fracture behavior gives rise to a phenomenon of higher harmonics during the wave transmission across the fracture. The higher harmonics contribute to the increase of jTnon j from jTlin j.

1. Introduction Extensive studies have been conducted to investigate the e¨ects of fractures on stress wave propagation in fractured media. Many of these studies have been concerned with the e¨ects of cracks with small size relative to a wavelength (i.e. micro-cracks). The theories of wave scattering at cracks have been developed with the considerations of linear contact conditions of crack faces (e.g., Achenbach, 1973; Hudson, 1980, 1981 and 1990; Angle and Achenbach, 1985a,b; Hirose and Kitahara, 1991; and many others), and nonlinear contact conditions of crack faces (e.g., Achenbach and Norris, 1982; Smyshlyaev and Willis, 1994; Capuani and Willis, 1997). Morris et al. (1979) and Hirose and Achenbach (1993) showed that the nonlinear contact conditions of cracks lead to a particular phenomenon of

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J. Zhao and J. G. Cai

high harmonics. In comparison with the micro-cracks, the macro-fractures (i.e. rock joints) may be more dominant in a fractured rock mass on most occasions. A planar macro-fracture, which is assumed to be large in extent and very thin in thickness relative to a wavelength (hereafter termed simply as fracture), can be physically viewed as a planar collection of collinear micro-voids and asperities in contact. The closure of voids and the deformation of asperities make the fracture compliant, and produce the overall deformation of the fracture. When waves propagate across such a fracture, the stress ®eld is continuous, but the displacement ®eld is discontinuous due to the fracture deformation. To account for the e¨ects of fractures on wave propagation, displacement discontinuity theories (or termed as non-welded or slip interface theories by some researchers) have been developed by treating the fracture deformational behaviors as displacement discontinuity boundary conditions in the wave equation. Di¨erent considerations of fracture deformational behaviors generate various displacement discontinuity models. Linear elastic displacement discontinuity models (spring models) for dry fractures have been established by Mindlin (1960), Jones (1967), Kendall and Tabor (1971), Schoenberg (1980 and 1983) and Kitsunezaki (1983). The linear models have been further developed and veri®ed through laboratory ultrasonic tests on arti®cial and natural rock fractures by Myer et al. (1985 and 1990), PyrakNolte (1988 and 1996), Pyrak-Nolte et al. (1987 and 1990a,b), Gu et al. (1996) and Roy and Pyrak-Nolte (1995 and 1997), as well as through ®eld seismic crosshole tests in a fractured rock mass by King et al. (1986) and Myer et al. (1995). Daehnke and Rossmanith (1997a) related the re¯ected and transmitted wave amplitudes across a linear elastic displacement discontinuity to the actual rock fracture surface properties. With the linear model established, analytical solutions of re¯ection and transmission coe½cients for arbitrarily incident waves transmitting across single fractures have been derived (Schoenberg, 1980; Pyrak-Nolte, 1990a). The linear elastic displacement discontinuity models for dry fractures are valid, provided that the magnitude of the seismic stress is insu½cient to mobilize the nonlinear asperity deformation and frictional slip of the fracture (Yi et al., 1997). This situation is typical in engineering practices such as ultrasonic detection of cracks in materials and most seismic investigations in fractured rock masses, where the wave amplitudes are relatively small. However, it has been found that the complete deformational (either normal or shear) behaviors of rock fractures are generally nonlinear (e.g., Kulhaway, 1975; Goodman, 1976; Bandis et al., 1983; Barton et al., 1985). In theory, linearity can be treated as a special case of nonlinearity of deformational behaviors of the fractures. To produce a general methodology, it is necessary to establish nonlinear displacement discontinuity models. In practice, there is a requirement for considering the nonlinear deformational behavior of the fractures in solving the problems of large amplitude wave propagation, i.e. blasting wave propagation from the sources of explosion, blasting and rockburst in underground excavation and mining. E¨orts have been made to establish the nonlinear displacement discontinuity models based on the slip rate-dependent frictional (shear) behavior of rock fractures. For example, Miller (1977 and 1978) presented theoretical studies into nor-

Transmission of P-waves across Single Fractures with Nonlinear Behavior

5

mally incident S-wave transmission across single fractures, where the shear stress was assumed to be a nonlinear function of frictional slip and sip rate of the fracture. Due to the mathematical di½culties in handling the nonlinear equations for solving the re¯ection and transmission coe½cients, an approximate analytical approach was adopted in Miller's studies to simplify the nonlinearity into linearity of the fracture shear behavior. His results showed that the wave transmission coe½cient depends on the ratio of the incident stress wave amplitude to a characteristic frictional stress associated with the nonlinear model. Chen et al. (1993) made laboratory measurements of the amplitudes of normally incident shear waves transmitting across a natural rock fracture during forced frictional slip. A distinct di¨erence in the transmitted amplitude between stable-slip and stick-slip has been observed. The purpose of this paper is to present a theoretical study on normally incident P-wave transmission across single dry fractures with a nonlinear normal deformational behavior. For this normally incident P-wave transmission, the shear deformation is not involved. Thus, the e¨ects of nonlinear normal deformation of fractures can be concentrated on. A nonlinear displacement discontinuity model is established by considering a nonlinear normal deformational behavior of fractures ± the hyperbolic elastic model (BB model), which is widely used to describe the nonlinear normal deformational behavior of rock fractures. In the theoretical derivation, no simpli®cation of nonlinearity into linearity of fracture deformational behavior is made. Parametric studies are conducted to obtain an insight into the e¨ects of the fracture nonlinear deformation on wave transmission, in terms of initial normal sti¨ness and the ratio of current maximum closure to maximum allowable closure of the fractures, as well as the incident wave amplitude and frequency. The phenomena of higher harmonics generated due to the fracture nonlinear deformational behavior are examined. Finally, the di¨erence and linkage between the wave transmission solutions for the linear and nonlinear displacement discontinuity models are discussed and explained.

2. Theoretical Formulations The normal deformational behaviour of rock fractures has been widely investigated under quasistatic or cyclic loading conditions (e.g., Goodman, 1976; Bandis et al., 1983; Barton et al., 1985; Zhao and Brown, 1992; and others). Of them, the hyperbolic model (BB model) proposed for matched fractures by Bandis et al. (1983 and 1985) is commonly used in rock mechanics and engineering. The BB model was established on the basis of a large number of laboratory experiments on matched fractures, where a hyperbolic function was used to express the relation of normal e¨ective stress vs fracture closure. Under cyclic loading/unloading conditions, the BB model describes that the initial loading and unloading cycles may cause a hysteresis between them. Successive loading/unloading cycles can continue to sti¨en the fractures, and the BB model eventually tends to a hyperbolic elastic model without the hysteresis between the load and unload cycles. In the present study, the hyperbolic elastic model is adopted. This is reasonable for the in situ

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J. Zhao and J. G. Cai

Fig. 1. Illustration of the hyperbolic elastic model (BB model) and linear elastic model of normal deformational behavior of a fracture

rock fractures, which have experienced multiple deformations in geological histories. No intrinsic attenuation mechanisms will occur during wave transmission across the fractures described by the hyperbolic elastic model. If the fracture closure (opening) and compression (tension) are assumed to be positive (negative), the normal e¨ective stress-closure relation is sn ; …1† dn ˆ kni ‡ …sn =dma † where dn is the fracture closure, sn is the normal e¨ective stress, dma is the maximum allowable closure of the fracture, and, kni is the normal sti¨ness of the fracture at initial stress. Figure 1 schematically illustrates a typical curve of the above hyperbolic relation, in comparison with the linear relation. Contrary to the linear model, the hyperbolic model involves fracture sti¨ness that increases with the increasing normal e¨ective stress. When the normal stress increases nearly to an in®nite value …sn ! y†, the fracture closure approaches the maximum allowable closure …dn ! dma †. The values of dma and kni can be determined from the laboratory measurements on fracture roughness coe½cient (JRC), fracture surface compressive strength (JCS) and average aperture thickness (aj †, as described by Bandis et al. (1983 and 1985). Now suppose that there is a fracture at x ˆ x1 in a half-space of a linear elastic, homogeneous and isotropic rock, as shown in Fig. 2a. The half-space is initially undisturbed. The left boundary of the half-space at x ˆ 0 is subjected to

Transmission of P-waves across Single Fractures with Nonlinear Behavior

7

Fig. 2. (a) Diagram of normally incident P-wave transmission across a fracture in a half-space, and (b) the corresponding left- and right-running characteristics in the x ÿ t plane

a normally incident planar P-wave, which is expressed as time histories of particle velocity …p…t††. During the wave transmission across the fracture, the displacement discontinuity boundary conditions of the fracture at x ˆ x1 are de®ned by the BB model of fracture normal behavior, sÿ …x1 ; t† ˆ s‡ …x1 ; t†; uÿ …x1 ; t† ÿ u‡ …x1 ; t† ˆ

s…x1 ; t† ; kni ‡ …s…x1 ; t†=dma †

…2† …3†

where s is the seismic stress normal to fracture, u is the seismic displacement normal to fracture, superscripts …ÿ† and …‡† refer to the wave ®elds before and after the fracture, and, kni and dma are the same as de®ned for Eq. (1).

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J. Zhao and J. G. Cai

The derivative expression of Eq. (3) is vÿ …x1 ; t† ÿ v‡ …x1 ; t† ˆ

qs…x1 ; t†  qt

1 qs…x1 ; t† s…x1 ; t† ÿ  ; s…x1 ; t† qt s…x1 ; t† 2 kni ‡ dma kni ‡ dma dma …4†

where vÿ …x1 ; t† and v‡ …x1 ; t† are the particle velocities in the wave ®elds before and after the fracture, respectively. (Note: vÿ …x1 ; t† is the particle velocity superposed by the incident and re¯ected waves, and v‡ …x1 ; t† is the particle velocity of the transmitted wave across the fracture). Although the displacement discontinuity boundary conditions are nonlinear as seen in Eqs. (2±4), the wave ®elds before and after the fracture are linearly elastic, where the method of characteristics is applicable. The method of characteristics has been widely used for solving the problems of one-dimensional wave propagation in continuous linearly elastic media (e.g., Ewing et al., 1957; Bedford et al., 1994). Cai and Zhao (2000) have con®rmed the applicability of the method for one-dimensional elastic wave re¯ections and transmissions normal to multiple parallel fractures. The method of characteristics derives that, the quantity zv…x; t† ‡ s…x; t† ˆ constant

…5†

along any straight line with slope a in the x ÿ t plane for one-dimensional wave propagation along x-axis in a linear elastic medium. Similarly, the quantity zv…x; t† ÿ s…x; t† ˆ constant

…6†

along any straight line with slope ÿa in the x ÿ t plane. In the equations, z is the seismic impedance, a is the wave propagation velocity, v…x; t† is the particle velocity and s…x; t† is the seismic stress. It should be noted that in these equations, s is de®ned to be positive for compression and negative for tension. This is for consistency with the common de®nition used in rock mechanics. The straight lines with slope a and ÿa in the x ÿ t plane are termed as rightrunning and left-running characteristics of the one-dimensional wave equation. As shown in Fig. 2b, a left-running characteristic ab is drawn, within the wave ®eld after the fracture, from the point …x1 ; t† back to the x-axis. Along the left-running characteristic ab, the quantity, zv‡ …x; t† ÿ s…x; t†, is a constant, where z is the seismic impedance of rock material. Since the half-space is undisturbed at t ˆ 0, the particle velocity v‡ …x; 0† ˆ 0 and s…x; 0† ˆ 0 at each point on the x-axis. Therefore, along line ab, zv‡ …x1 ; t† ÿ s…x1 ; t† ˆ 0:

…7†

Similarly, a right-running characteristic ac is drawn, within the wave ®eld before the fracture, from the point …x1 ; t† back to the t-axis. It intersects the t-axis at the point …0; t ÿ x1 =a†, where vÿ …0; t ÿ x1 =a† is equal to p…t ÿ x1 =a† (particle velocity input to the left boundary at time t ÿ x1 =a†. Therefore, along the right-running

Transmission of P-waves across Single Fractures with Nonlinear Behavior

9

characteristic ac, zvÿ …x1 ; t† ‡ s…x1 ; t† ˆ zp…t ÿ x1 =a† ‡ s…0; t ÿ x1 =a†;

…8†

where a is the P-wave propagation velocity. Another left-running characteristic cd is drawn, within the wave ®eld before the fracture, from point …0; t ÿ x1 =a† back to the x-axis. Along the left-running characteristic cd, zp…t ÿ x1 =a† ÿ s…0; t ÿ x1 =a† ˆ 0:

…9†

Based on Eq. (9), s…0; t ÿ x1 =a† is determined. Hence, Eq. (8) becomes zvÿ …x1 ; t† ‡ s…x1 ; t† ˆ 2zp…t ÿ x1 =a†:

…10†

Summing up Eqs. (7) and (10) yields vÿ …x1 ; t† ‡ v‡ …x1 ; t† ˆ 2p…t ÿ x1 =a†:

…11†

Equation (11) expresses the relation between particle velocities before and after the fracture. According to Eq. (7), s…x1 ; t† is equal to zv‡ …x1 ; t†. So, Eq. (4) becomes vÿ …x1 ; t† ÿ v‡ …x1 ; t† ˆ

zqv‡ …x1 ; t† 1   zv‡ …x1 ; t† qt kni ‡ dma ÿ

zqv‡ …x1 ; t† qt

 dma

zv‡ …x1 ; t† zv‡ …x1 ; t† kni ‡ dma

2 :

…12†

Based on Eqs. (11) and (12), a di¨erential equation with respect to v‡ …x1 ; t† can be derived: qv‡ …x1 ; t† ˆ8 qt >
z z 2 v‡ …x1 ; t† =  ÿ  2 zv…x1 ; t† zv…x ; t† 1 > > dm kni ‡ ; : kni ‡ d ma dma

…13†

However, the closed-form solution for v‡ …x1 ; t† is di½cult to obtain directly from Eq. (13). In the following section, this equation is numerically solved for v‡ …x1 ; t†.

3. Numeric Solutions In order to obtain the numeric solution for v‡ …x1 ; t†, the time interval ‰0; tŠ is divided into J equal time steps, namely, ‰0; tŠ ˆ ‰0; t1 ; . . . tj . . . ; tJ Š … j ˆ 1; 2; . . . ; J†.

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J. Zhao and J. G. Cai

Within each time step, a linear approximation of

qv‡ …x1 ; t† is adopted: qt

qv‡ …x1 ; tj † v‡ …x1 ; tj‡1 † ÿ v‡ …x1 ; tj † ˆ : qt Dt

…14†

Te , m where Te is the period of the incident wave, and m is the number (integer) of time steps within one period of the incident wave. Thus, Eq. (14) can be expressed as: The time increment …Dt† for each time step can always be selected as Dt ˆ

qv‡ …x1 ; tj † m…v‡ …x1 ; tj‡1 † ÿ v‡ …x1 ; tj †† ˆ : qt Te

…15†

Therefore, Eq. (13) becomes v‡ …x1 ; tj‡1 † ˆ

Te…2p…t ÿ x1 =a† ÿ 2v‡ …x1 ; tj †† 9 ‡ v‡ …x1 ; tj †: 8 2 ‡ > > z v …x ; t † z 1 j = > dma kni ‡ ; : kni ‡ dma dma …16† ‡

Equation (16) is a recurrence equation for calculating v …x1 ; t†. When the wave input, p…t†, and the initial particle velocity condition, v‡ …x1 ; 0†, are speci®ed, v‡ …x1 ; t† can be determined in an iterative calculating process. In the calculation, if m is selected to be large enough, a su½ciently small Dt can be obtained. Therefore, numeric results of v‡ …x1 ; t† with su½cient accuracy can always be gained. If Te is replaced with Te ˆ 1=f in Eq. (16), it is obvious that the calculation of v‡ …x1 ; t† depends on the incident wave frequency … f †. From v‡ …x1 ; t†, vÿ …x1 ; t† can be calculated according to Eq. (11). Thus, the particle velocity re¯ected o¨ the fracture can be calculated by vref …x1 ; t† ˆ vÿ …x1 ; t† ÿ vinc …x1 ; t†, where vinc …x1 ; t† is the time histories of incident particle velocity.

4. Parametric Studies Parametric studies are conducted to obtain an insight into the e¨ects of the nonlinear fracture deformation on wave transmission, in terms of initial normal sti¨ness and the ratio of maximum closure to maximum allowable closure of the fractures, as well as the incident wave amplitude and frequency. In numeric calculations, the rock density …r† is set to 2:4  10 3 kg/m 3 , and the P-wave propagation velocity …a† is 4500 m/s. Selecting the number …m† of time steps within one period wave is through trial and error. It is found that  …Te† of the incident  Te Te > may lead to an unstable di¨erential calculation with the m < 100 Dt ˆ m 100

Transmission of P-waves across Single Fractures with Nonlinear Behavior

11

linear approximation, if the amplitude of input wave …vinc …t†† is a highly nonlinear function of the time t. In this parametric study, the particle velocity input applied normally to the left boundary …x ˆ 0† is sinusoidal waves in the form of vinc …t† ˆ Vinc sin ot, where Vinc is the amplitude in particle velocity and o is the angular frequency. For the sinusoidal form of wave input, m is set to be a large number of 500 to ensure a stable di¨erential calculation. (This has been con®rmed by the later calculation). The parametric studies are performed ®rstly for the incident waves with different amplitudes. For example, the particle velocity amplitude …Vinc † is set to 0.02, 0.05, 0.10, 0.13 and 0.20 m/s, respectively. The corresponding stress amplitude …sinc ˆ raVinc † of the incident waves is 0.22, 0.54, 1.1, 1.4 and 2.2 MPa. The wave frequency is set to 50 Hz. The fracture has the initial normal sti¨ness of 1250 MPa/m and the maximum allowable closure of 0.61 mm. Calculating results of time histories of v‡ …x1 ; t† are shown together with the corresponding vinc …x1 ; t† (here is a half sinusoidal pulse) in the left column of Fig. 3. The nonlinear closure of the fracture subjected to seismic stresses induced by the incident waves …vinc …x1 ; t†† with di¨erent amplitudes …Vinc † are shown in the right column of Fig. 3. As shown, during the transmission of a P-wave pulse, the fracture is compressed and then relaxed to the initial state. In other words, the fracture closure increases to a current maximum value …dcm † and then decreases to zero in a nonlinear mode. During this process, the magnitude of transmission coe½cient …jTnon j† is calculated by the ratio of the resulting transmitted amplitude …Vtra ˆ peak value of v‡ …x1 ; t†† to the incident amplitude …Vinc †. The results of jTnon j are shown in Fig. 3 for each case of incident amplitudes. It can be seen that an increase in the incident wave amplitude correspondingly causes an increase in fracture current maximum closure …dcm † during the wave transmission, and in turn, the increased ratio …1† of dcm to maximum allowable closure …dma †. As a ®nal result, jTnon j is increased as well. Further calculations of jTnon j at f ˆ 50 Hz are carried out for more magnitude levels of amplitude and various combinations between kni and dma . In the calculations, the particle velocity amplitude …Vinc † is varied within a wide range, i.e. Vinc ˆ 0:01 ÿ 1:0 m/s …sinc ˆ 0:11 ÿ 11:0 MPa). The purpose is to fully demonstrate the mobilization of nonlinearity in fracture deformation, and the variation of transmission coe½cient with wave amplitudes. The combinations between kni and dma are determined from various values of JRC and JCS of the fracture, according to the empirical formulas given by Bandis et al. (1983). Figure 4a shows jTnon j as a function of 1 for various combinations between kni and dma . As can be seen, jTnon j increases with the increased value of kni . This implies that an initially sti¨er fracture produces more wave transmission across the fracture. On the other hand, jTnon j increases with the decreased value of dma , indicating that the fracture with smaller aperture generates more wave transmission across the fracture. Figure 4a shows that jTnon j consistently increases with the increased value of 1. For the purpose of comparisons, the transmission coe½cients …jTlin j† for a linear deformable fracture with the constant sti¨ness kn ˆ kni are also plotted in Fig. 4a. The transmission coe½cient …jTlin j† for wave incidence normal to a linear deformable fracture in identical rocks is calculated according to the analytical

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J. Zhao and J. G. Cai

Fig. 3. Incident P-wave pulses with di¨erent amplitudes …Vinc † and the resulting transmitted pulses …Vtra † across a fracture, and the corresponding nonlinear closure curve of the fracture … f ˆ 50 Hz, kni ˆ 1:25 GPa/m and dma ˆ 0:61 mm)

Transmission of P-waves across Single Fractures with Nonlinear Behavior

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Fig. 3. (continued)

solution obtained by Schoenberg (1980) and Pyrak-Nolte (1990a). It is   k 2 zo  ; Tlin ˆ k ÿi ‡ 2 zo 2  2 31=2 k 6 4 zo 7 6 7 …for compressional and shear waves†; jTlin j ˆ 6  2 7 4 5 k 4 ‡1 zo

…17†

where o is the angular wave frequency. For compressional wave incidence, k denotes the fracture normal sti¨ness, and z…ˆ rap † the seismic impedance of the

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J. Zhao and J. G. Cai

a)

b)

Fig. 4. jTnon j as a function of g for various combinations between kni and dma in comparison with jTlin j. (a) f ˆ 50 Hz, (b) f ˆ 150 Hz

rock for compressional waves. For shear wave incidence, k denotes the shear sti¨ness, and z…ˆ ras † the seismic impedance for shear waves. Equation (17) shows that Tlin depends on the wave frequency, the fracture sti¨ness and the seismic impedance of rock. As shown, jTlin j is independent of 1, while jTnon j is strongly dependent on 1. jTnon j increases from jTlin j to 1, depending on the value of 1. If 1 is very small, jTnon j is nearly equal to jTlin j for a linear deformable fracture with the same sti¨ness as the nonlinear initial fracture sti¨ness. If 1 becomes bigger, jTnon j increases from jTlin j. Furthermore, if 1 is big enough (i.e. 1 V 0:8 as shown in Fig. 4), jTnon j approaches 1. The variation of jTnon j with 1 (as a nondimensional parameter) re¯ects the combined e¨ects of Vinc and dma on wave transmission. For example, an increased Vinc or a decreased dma can result in an increase in the ratio …1† of dcm to dma (the fracture is tightly compressed), and in turn, an increase of jTnon j. Moreover, by increasing the incident wave frequency … f †, jTnon j is calculated and compared with the corresponding jTlin j. Generally, jTnon j decreases with the increased values of f. As an example, Fig. 4b shows jTnon j as a function of 1 at f ˆ 150 Hz. As can be seen, jTnon j at f ˆ 150 Hz is lower than that at f ˆ 50 Hz and has the similar trend of variation with kni , dma and 1.

Transmission of P-waves across Single Fractures with Nonlinear Behavior

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In addition to jTnon j, the magnitude of re¯ection coe½cient …jRnon j† is calculated from the ratio of the re¯ected amplitude …Vref † to the incident amplitude …Vinc †. jRnon j for f ˆ 50 Hz and 150 Hz is shown respectively in Figs. 5a and 5b, together with the re¯ection coe½cient …jRlin j† for a linear deformable fracture with the constant sti¨ness kn ˆ kni . The re¯ection coe½cient …jTlin j† for P-wave incidence normal to a linear deformable fracture in identical rocks is calculated according to the analytical solution obtained by Schoenberg (1980) and Pyrak-Nolte (1990a). It is, Rlin ˆ

i  ; k ÿi ‡ 2 zo

31=2 1 6  2 7 : jRlin j ˆ4 5 k ‡1 4 zo

…18†

2

…for compressional wave incidence†;

where o, k and z have the same de®nition as in Eq. (17). As shown in Figs. 5a and 5b, jRnon j always has the variation trend converse to jTnon j, i.e. jRnon j decreases from jRlin j with the increased values of kni and 1 as well as the decreased values of dma and f . In order to check whether the incident wave energy is preserved during the wave transmission and re¯ection at the fracture, transmitted and re¯ected energy rates are calculated by numeric integration of v‡ …x1 ; t† and vref …x1 ; t†: 0 0 „ ttra P jˆttra ‡Ttra ‡Ttra z…v‡ …x1 ; t†† 2 dt z…v‡ …x1 ; tj †† 2 Dt 0 0 Etra ttra jˆttra ˆ 0 ‡T ˆ P jˆt 0 ‡T ; …19† etra ˆ inc inc Einc „ tinc inc z…vinc …x1 ; t†† 2 dt z…vinc …x1 ; tj †† 2 Dt t0 jˆt 0 inc

eref

inc

0 0 „ tref P jˆtref ‡Tref ‡Tref z…vref …x1 ; t†† 2 dt z…vref …x1 ; tj †† 2 Dt 0 0 Eref tref jˆtref ˆ P jˆt 0 ‡T ; ˆ ˆ 0 ‡T inc inc Einc „ tinc inc z…v …x ; t†† 2 dt z…v …x ; t †† 2 Dt 0 tinc

inc

1

0 jˆtinc

inc

1

…20†

j

where etra , eref are the energy rates of transmitted and re¯ected waves respectively, Etra , Eref , Einc are the energies of transmitted, re¯ected and incident waves, Ttra , Tref , Tinc are the periods of transmitted, re¯ected and incident waves, 0 0 0 and, ttra , tref , tinc are the initial time of transmitted, re¯ected and incident waves. The results show that, etra …eref † increases with the increased jTnon j …jRnon j†, and eref ‡ etra ˆ 1 for all the calculating cases. This means that, the balance of energy rates of transmitted and re¯ected waves is always preserved for waves propagating across the nonlinear elastic deformable fracture. (Because these results are obtained by numeric integration, eref ‡ etra may not be exactly equal to 1, i.e. equal to 0.985 or 1.015).

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J. Zhao and J. G. Cai

a)

b)

Fig. 5. jRnon j as a function of g for various combinations between kni and dma in comparison with jRlin j. (a) f ˆ 50 Hz, (b) f ˆ 150 Hz

For a nonlinear deformable fracture, jTnon j ˆ 1 represents that the transmitted wave amplitude is equal to the incident wave amplitude. In such a situation, however, the transmitted wave energy may be less than the incident wave energy. This is indicated by the results that when jTnon j ˆ 1, etra U 1 and eref V 0. Another indication is that, when jTnon j ˆ 1, there are still waves re¯ected o¨ the fracture and jRnon j has a limited value as shown in Figs. 4 and 5. In order to examine the frequency components of the transmitted waves, the calculating results of v‡ …x1 ; t† are analyzed in the frequency domain with the FFT algorithm. Higher harmonics are found from the frequency spectra of v‡ …x1 ; t†, which are indicated by the components of higher frequencies multiple times the fundamental-order frequency of the incident waves. This phenomenon is more signi®cant when Vinc and 1 are relatively large and kni is small, because under these conditions the fracture has a relatively more complete hyperbolic deformational behavior. Figure 6 illustrates a typical example of transmitted waves …v‡ …x1 ; t†† across a fracture of kni ˆ 1250 MPa/m (a relatively small value) and dma ˆ 0:61 mm, resulting from the incident sinusoidal waves …vinc …x1 ; t†† of Vinc ˆ 0:3 m/s. The corresponding g is equal to 0.97 (a large value). As can be seen, the transmitted waves …v‡ …x1 ; t†† have the same amplitude (a complete amplitude trans-

Transmission of P-waves across Single Fractures with Nonlinear Behavior

17

Fig. 6. Examples of incident waves and the resulting transmitted waves across a fracture with the hyperbolic deformational behavior …Vinc ˆ 0:3 m/s, f ˆ 50 Hz, kni ˆ 1:25 GPa/m, dma ˆ 0:61 mm and g ˆ 0:98†

mission case) but a distorted waveform, compared to the smooth waveform of the incident sinusoidal wave …vinc …x1 ; t††. The frequency spectra of incident and transmitted waves are compared in Fig. 7. As shown, the incident waves have only one frequency component at f ˆ 50 Hz (termed as fundamental component), while the transmitted waves involve not only the fundamental component … f ˆ 50 Hz) but also higher frequency components at f ˆ …2n ÿ 1†  50 Hz …n ˆ 1; 2; 3 . . .†. These higher components are called higher harmonics. The Fourier amplitudes of the higher harmonics gradually become weaker in the sequence of

Fig. 7. Comparison of frequency spectra of the incident and transmitted waves that are shown in Fig. 6

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J. Zhao and J. G. Cai

Fig. 8. jTnon j for the fundamental, the second and the third harmonics as a function of g

f ˆ 50, 150, 250 and 350 Hz until being negligible in amplitude. According to the fourier amplitudes, the magnitudes of transmission coe½cients for the fundamental and higher harmonics can be calculated. Thus, the overall jTnon j of a nonlinear fracture can be viewed to be composed of those for fundamental and higher harmonics. Figure 8 shows the ``decomposed'' jTnon j individually for the fundamental, second and third order of harmonics as a function of 1, where the incident wave frequency f ˆ 50 Hz, kni ˆ 1250 MPa/m and dma ˆ 0:61 mm, but Vinc is varied from 0.02 to 0.3 m/s. As seen from this ®gure, the magnitudes of transmission coe½cients for the fundamental, second and third harmonics increase with the increasing value of 1. If the value of 1 is very small, the magnitude of transmission coe½cient for the fundamental harmonics is equal to jTlin j for a linear deformable fracture, and the higher harmonics do not occur. This indicates that, under the condition of a small 1, the boundary-type nonlinear problem of wave transmission can be equivalently treated as a linear problem. 5. Discussions and Conclusions In this study the linear displacement discontinuity model is extended to a nonlinear model ± the hyperbolic elastic model (BB model), where the method of characteristics is used. This method is simple and e¨ective in deriving the solutions of particle velocities in the wave ®elds before and after the fracture. In the nonlinear model established, the P-wave transmission amplitude across the fracture is related to the nonlinear closure behavior of the fracture. Compared to a linear

Transmission of P-waves across Single Fractures with Nonlinear Behavior

19

deformable fracture, a nonlinear deformable fracture can generate more wave transmission (less wave attenuation) in amplitudes and higher harmonics in frequencies. The magnitude of transmission coe½cient …jTnon j† for a nonlinear deformable fracture depends not only on the fracture sti¨ness and wave frequency (like the case for a linear deformable fracture), but also on the ratio …1† of current maximum closure …dcm † to maximum allowable closure …dma † of the fracture. In the parametric studies, the incident wave amplitude …Vinc † in particle velocity is varied from 0.01 to 1.0 m/s. Vinc ˆ 0:01 m/s (sinc ˆ 0:11 MPa, corresponding to seismic waves in site investigation) is insu½cient to mobilize the deformational nonlinearity of fracture with kni ˆ 1:25 ÿ 5:5 GPa. On the other hand, Vinc ˆ 1:0 m/s …sinc ˆ 11:0 MPa, corresponding to blasting waves at a certain distance from blasting center) is su½cient to mobilize the deformational nonlinearity of fracture with kni ˆ 1:25 ÿ 5:5 GPa. Therefore, a signi®cant variation of jTnon j with 1 is illustrated. It is shown that, if a nonlinear deformable fracture is slightly compressed by small amplitude waves, it acts like a linear fracture, and there is relatively less wave transmission (more wave attenuation) across the fracture. Whereas, if the fracture is tightly compressed by larger amplitude waves, there is more wave transmission (less wave attenuation). At a limiting case, if the fracture is compressed close to the allowable extreme (the complete closure of fracture), there is complete wave transmission (no wave attenuation) in amplitudes. In the frequency domain, the transmitted waves across the nonlinear deformable fracture involve not only the fundamental frequency component of the incident waves, but also higher frequency components multiple times the fundamental component. This is di¨erent from the common knowledge for the case of wave transmission across linear deformable fractures, where the high frequency components of the incident waves are ®ltered out in the transmitted waves across the fractures. The higher harmonics are a special phenomenon generated due to the nonlinear deformational behavior of fractures. However, the physical mechanism of causing the high harmonics is not well understood. A possible explanation could be stated as follows. The nonlinear deformable fracture, which is sti¨ened in a time-dependent nonlinear process by the external applied stress waves, can transmit more energy than the linear deformable fracture. During this nonlinear process, the extra energy tends to get released in the forms of high frequency waves. This special phenomenon may be useful for detecting fractures and for determining deformational properties of fractures in rock masses. The mechanisms of jTnon j increasing with the increased value of 1 can be attributed to the increasing fracture sti¨ness and higher harmonics during the wave propagation across the nonlinear deformable fracture, If 1 is small, the fracture sti¨ness is nearly constant at the initial sti¨ness, and no higher harmonics occur. Under this situation, the nonlinear fracture behaves like a linear fracture, and jTnon j is therefore equal to jTlin j. If 1 is big, the fracture sti¨ness increases from the initial sti¨ness in a nonlinear mode. During this process, higher harmonic waves are generated and transmitted across the fracture, in addition to the fundamental frequency wave. This results in an increase in jTnon j. Althought this study falls into the special case of normal incidence of P-waves,

20

J. Zhao and J. G. Cai

it provides an insight into the di¨erence and linkage between the e¨ects of linear and nonlinear fracture normal deformations on elastic wave transmission. The further work is to apply the nonlinear approach to solve the problem of onedimensional S-wave transmission normal to fractures with a nonlinear shear behavior. While for the cases of wave incidence non-normal to fractures, the method of characteristics is not applicable, because those cases are two-dimensional problems and wave conversion occurs.

References Achenbach, J. D. (1973): Wave propagation in elastic solids. North Holland, Amsterdam. Achenbach, J. D., Norris, A. N. (1982): Loss of specular re¯ection due to nonlinear crackface interaction. J. Nondestruct. Eval. 3 (4), 229±239. Angle, Y. C., Achenbach, J. D. (1985a): Re¯ection and transmission of elastic waves by a periodic array of cracks. J. Appl. Mech. 52, 33. Angle, Y. C., Achenbach, J. D. (1985b): Re¯ection and transmission of elastic waves by a periodic array of cracks: oblique incidence. Wave Motion 9, 375. Bandis, S. C., Lumsden, A. C., Barton, N. R. (1983): Fundamentals of rock fracture deformation. Int. J. Rock Mech. Min. Sci Geomech. Abstr. 20 (6), 249±268. Bandis, S. C., Barton, N. R., Christianson, M. (1985): Application of a new numerical model of joint behavior to rock mechanics problems. In: Proc., Int. Symposium on Fundamentals of Rock Joints. Lulea, Sweden, 345±356. Bedford, A., Drumheller, D. S. (1994): Introduction to elastic wave propagation. Wiley & Sons, Chichester. Cai, J. G., Zhao, J. (2000): E¨ects of multiple parallel fractures on apparent attenuation of waves in rock masses. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 37(4), 661±682. Capuani, D., Willis, J. R. (1997): Wave propagation in elastic media with cracks Part I: transient nonlinear response of a single crack. Eur. J. Mech. A/Solids 16 (3), 377±408. Chen, W. Y., Lovell, C. W., Haley, G. M., Pyrak-Nolte, L. J. (1993): Variation of shear wave amplitude during frictional sliding. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 30 (7), 779±784. Daehnke, A., Rossmanith, H. P. (1997a): Re¯ection and transmission of plane stress waves at interfaces modeling various rock joints. Int. J. Blasting Fragment. 1 (2), 111±231. Daehnke, A., Rossmanith, H. P. (1997b): Re¯ection and refraction of plane stress waves at dissipative interfaces. In: Proc., 1st Int. Conference on Damage and Failure of Interfaces. Vienna, Austria, 315±320. Ewing, W. M., Jardetzky, W. S., Press, F. (1957): Elastic waves in layered media. McGrawHill, New York. Goodman, R. E. (1976): Methods of geological engineering in discontinuous rocks, 1st edn. West, New York. Gu, B., SuaÂrez-Rivera, R., Nihei, K., Myer, L. R. (1996): Incidence of plane waves upon a fracture. J. Geophys. Res. 101 (B11), 25337±25346. Hirose, S., Achenbach, J. D. (1993): Higher harmonics in the far ®eld due to dynamic crack-face contacting. J. Acoustic Soc. Am. 93 (1), 142±147.

Transmission of P-waves across Single Fractures with Nonlinear Behavior

21

Hirose, S., Kitahara, M. (1991): Scattering of elastic waves by a crack with spring-mass contact. Int. J. Numer. Meth. Engng. 31, 789±801. Hudson, J. A. (1980): Overall properties of a cracked solid. Math. Proc. Camb. Phil. Soc. 88, 371±384. Hudson, J. A. (1981): Wave speeds and attenuation of elastic waves in material containing cracks. Geophys. J. Roy. Astronom. Soc. 64, 133±150. Hudson, J. A. (1990): Attenuation due to second-order scattering in material containing cracks. Geophys. J. Int. 102, 485±490. Jones, J. P., Whittier, J. S. (1967): Waves at a ¯exibly bonded interface. J. Appl. Mech. 40, 905±909. Kendall, K., Tabor, D. (1971): An ultrasonic study of the area of contact between stationary and sliding surfaces. Proc. Roy. Soc. London A 323, 321. King, M. S., Myer, L. R., Rezowalli, J. J. (1986): Experimental studies of elastic wave propagation in a columnar jointed rock mass. Geophys. Prospect. 34 (8), 1185±1199. Kitsunezaki, C. (1983): Behavior of plane elastic waves across a plane crack. J. Mining College of Akita University A6 (3), 173±187. Kulhaway, F. H. (1975): Stress-deformation properties of rock and rock discontinuities. Engng. Geol. 8, 327±350. Miller, R. K. (1977): An approximate method of analysis of the transmission of elastic waves through a frictional boundary. J. Appl. Mech. 44, 652±656. Miller, R. K. (1978): The e¨ects of boundary friction on the propagation of elastic waves. Bull. Seismol. Soc. Am. 68 (4), 987±998. Mindlin, R. D. (1960): Waves and vibrations in isotropic elastic planes. In: Proc., 1st Symposium on Naval Structural Mechanics, Standford University, USA, 339. Morris, W. L., Buck, O., Inman, R. V. (1979): Acoustic harmonic generation due to fatigue damage in high-strength aluminum. J. Appl. Phys. 50, 6737±6741. Myer, L. R., Hopkins, D., Cook, N. G. W. (1985): E¨ects of contact area of an interface on acoustic wave transmission characteristics. In: Proc., 26th US Rock Mechanics Symposium, Boston, 1, 565±572. Myer, L. R., Pyrak-Nolte, L. J., Cook, N. G. W. (1990): E¨ects of single fracture on seismic wave propagation. In: Proc., ISRM Symposium on Rock Fractures, Leon, 467± 473. Myer, L. R., Hopkins, D., Peterson, J. E., Cook, N. G. W. (1995): Seismic wave propagation across multiple fractures. Fract. Joint. Rock Masses, 105±109. Pyrak-Nolte, L. J. (1988): Seismic visibility of fractures. Ph. D. Thesis. University of California, Berkeley, USA. Pyrak-Nolte, L. J. (1996): The seismic response of fractures and the interrelations among fracture properties. Int. J. Rock Mech. Min. Sci. 33 (8), 787±802. Pyrak-Nolte, L. J., Myer, L. R., Cook, N. G. W. (1987): Seismic visibility of fractures. In: Proc., 28th US Symposium on Rock Mechanics, 47±56. Pyrak-Nolte, L. J., Myer, L. R., Cook, N. G. W. (1990a): Transmission of seismic waves across single natural fractures. J. Geophys. Res. 95 (B6), 8617±8638. Pyrak-Nolte, L. J., Myer, L. R., Cook, N. G. W. (1990b): Anisotropy in seismic velocities and amplitudes from multiple parallel fractures. J. Geophys. Res. 95 (B7), 11345±11358.

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J. Zhao and J. G. Cai: Transmission of Elastic P-waves

Roy, S., Pyrak-Nolte, L. J. (1995): Interface waves propagating along tensile fractures in dolomite. Geophys. Res. Lett. 22 (20), 2773±2777. Roy, S., Pyrak-Nolte, L. J. (1997): Observation of a distinct compressional-mode interface wave on a single fracture. Geophys. Res. Lett. 24 (2), 173±176. Schoenberg, M. (1980): Elastic wave behavior across linear slip interfaces. J. Acoustic Soc. Am. 68 (5), 1516±1521. Schoenberg, M. (1983): Re¯ection of elastic waves from periodically strati®ed media with interfacial slip. Geophys. Prospect. 31, 265±292. Smyshlyaev, V. P., Willis, J. R. (1994): Linear and nonlinear scattering of elastic waves by microcracks. J. Mech. Phys. Solids 42 (4), 585±610. SuaÂrez-Rivera, R., Cook, N. G. W., Myer, L. R. (1992): Study on the transmission of shear waves across thin liquid ®lms and thin clay layers. In: Proc., 33rd U.S. Rock Mechanics Symposium, Balkema, Rotterdam, 937±946. Yi, W., Nihei, K. T., Rector, J. W., Nakagawa, S., Myer, L. R., Cook, N. G. W. (1997): Frequency-dependence seismic anisotropy in fractured rock. Int. J. Rock Mech. Min. Sci. 34 (3±4), Paper No. 349. Zhao, J., Brown, E. T. (1992): Logarithmic relation for joint normal deformation under e¨ective stress. In: Proc., Int. Symposium on Rock Mechanics ± Eurock'92, 69±74. Authors' address: J. G. Cai, Nanyang Technological University, School of Civil and Structural Engineering, Block N1, a1A-29, Singapore 639798.