Simulation of viscoelastic behavior of defected rock by using numerical manifold method

Abstract

Numerical simulations of longitudinal wave propagation in a rock bar with microcracks are conducted by using the numerical manifold method which has great advantages in the simulation of discontinuities. Firstly, validation of the numerical manifold method is carried out by simulations of a longitudinal stress wave propagating through intact and cracked rock bars. The behavior of the stress wave traveling in a one-dimensional rock bar with randomly distributed microcracks is subsequently studied. It is revealed that the highly defected rock bar has significant viscoelasticity to the stress wave propagation. Wave attenuation as well as time delay is affected by the length, quantity, specific stiffness of the distributed microcracks as well as the incident stress wave frequency. The storage and loss moduli of the defected rock are also affected by the microcrack properties; however, they are independent of incident stress wave frequency.

Keywords

stress wave propagation / defected rock / numerical manifold method / viscoelastic behavior / storage modulus / loss modulus

Introduction

Microdefects are one of the most distinguishing features of rock materials, which affect both the static and dynamic behavior of rock materials remarkably. These microdefects (i.e. pores, open cracks, veins, etc.) may stem from earth crust motion, weathering, high pore pressure, intrinsic inhomogeneous mixtures and minerals that form the rock mass. Due to the existence of various defects, rock materials behave viscoelastically when stress wave propagation is taken into consideration. It is significant to quantitatively reveal the effects of microdefects in rock materials on the rock viscoelasticity.

Viscoelastic behavior of defected rock has been realized by previous researchers. Ichikawa et al. [1] observed a three-dimensional texture of granitic rock by X-ray tomographic microscopy, and established a micro/macromodel of granite by a homogenization analysis method to simulate the viscoelastic behavior, which caused time-dependent relaxation phenomenon of the granitic rock. Pyrak-Nolte [2] and Cook [3] found that natural rock fractures possessed elastic as well as viscous coupling across the interfaces according to extensive laboratory tests of ultrasonic wave attenuation. A viscoelastic medium was suggested by Pyrak-Nolte [2] to reflect the frequency dependence and wave attenuation affected by the fractures. Li et al. [4] put forward an equivalent viscoelastic medium model to simulate the viscoelasticity of rock mass with parallel joints. These studies mainly focused on the stress wave behavior of a rock mass with few rock fractures or joints, and the effect of randomly distributed microcracks on the viscoelastic behavior of rock materials is not possible to be investigated by simple experimental setup and simplified analytical derivations.

Apart from analytical derivation and laboratory experiments, numerical modeling and simulation of defected rock material is an effective complement, or even irreplaceable in some cases. Sometimes analytical expressions for the description of elastic wave propagation in the presence of fractures are only available for relative simple cases [5] (e. g. single cracks or parallel fractures with simple geometries). Numerical simulations play an important role in study the insight mechanisms of stress wave propagation in defected rock material in a convenient and economical way, especially when the rock material is highly cracked or with large-scale inclusions.

The numerical manifold method (NMM) [6,7] provides a natural bridge between the continuum based and discrete element based numerical methods, in which mathematical covers and physical covers are incorporated to make the description of cracks arbitrary and much more convenient. The mathematical covers of NMM used in the present study are regular hexagons, and they are independent and completely cover the physical domain in an overlapped manner. The intersection of the mathematical covers and physical domain generates physical covers, which can be regarded as the subdivision of the mathematical covers by the physical domain. Finally, overlapped parts of these physical covers form the manifold elements, which can have an arbitrary geometry. The cracks in the NMM are treated as contacts of physical boundaries. The normal specific stiffness and tangential specific stiffness of the cracks are modeled by normal and tangential springs with certain specific stiffness of the contact respectively. And the tangential specific stiffness is designated 0.4 times of the normal specific stiffness in this study. Therefore, the crack model in the NMM is coincide with the assumption of the displacement discontinuity method (DDM) [8-10], whose cracks is treated as imperfectly welded interfaces (displacement discontinuous boundaries) across which stresses are continuous, however, the displacements before and after the interface are discontinuous. Since its applicability and the specific feature in modeling discontinuous rock material, it is employed in the present study to study stress wave propagation in the highly cracked rock material.

Numerical simulations of stress wave propagation in a one-dimensional defected rock bar are performed. The NMM is validated by studying longitudinal (or P-) wave propagation through intact and cracked rock bars. Stress wave passing through a heavily defected rock bar with numerous randomly distributed microcracks is modeled to quantitatively investigate the rock viscoelasticity in terms of crack length, quantity, and specific stiffness. The relation of the frequency dependent dynamic complex modulus which is the main index of rock viscoelasticity and the parameters of the microcracks (i.e. crack length, quantity, specific stiffness) are also discussed through a parametric study.

Validation of NMM modeling on stress wave propagation

Although the NMM has been extensively applied in the simulation of discontinuous deformation analysis of jointed rock mass, its application on stress wave propagation has been seldom reported. It is necessary to validate the suitability and accuracy of the NMM for the simulation of stress wave propagation and attenuation across the fractured rock.

NMM incorporates the implicit algorithm and establishes the simultaneous equilibrium equations by minimizing the total potential energy of the entire physical system. When a stress wave propagates in an elastic material without damping, the total energy (summation of the potential energy and kinetic energy) will not decay providing that the system has no energy exchange with the environment. Furthermore, if the stress wave is traveling in a one-dimensional bar (when the wave length is more than ten times of the transverse dimension of the bar, the geometrical dissipation is small enough to be ignored), the amplitude of the stress wave in the longitudinal direction will not attenuate. A test (case 1) is carried out to confirm the energy conservation and invariance of the wave amplitude when a P-wave propagates in a one-dimensional elastic bar without defects as plotted in Fig. 1(a). The NMM model is illustrated in Fig. 1(b). It is the advantage of the NMM that the mathematical covers do not need to conform to the boundaries of the physical domain. It has the flexibility to simulation different density microcracks in the rock bar using consistent a mathematical cover system. Detail description of NMM models and advantages can be found in [6,7].

In order to investigate the stress wave property exclusively, no gravity force is applied. The dimensions of the rock bar are 2 m of length and 0.05 m of width. A half sinusoidal impacting wave with a frequency of 2500 Hz is applied at the left side of the rock bar to introduce a 1 MPa in amplitude compression stress wave. A measuring point (point A) is located at 1.0 m away from the left side of the bar as seen in Fig. 1(a). Other parameters are E = 75 GPa, ν = 0.25, ρ = 2680.98 kg/m³. The result of the test is presented in Fig. 2, from which three stress waves passing through point A are recorded, they are subsequently the incident wave, the reflected wave from the free end, and the re-reflected wave from the impact end. The amplitudes of the three stress waves are -0.999, 0.995 and -0.995 MPa (minus represents compression) respectively at point A, which demonstrates that the NMM model has excellent energy conservative property and accuracy in simulating stress wave propagation. The phase velocity of the stress wave is also obtained based on the time difference of the three peaks, and it is 5281.23 m/s (the error is -0.15% compared with the theoretical result, which is 5289.13 m/s). This test shows NMM has the ability in simulation of stress wave propagation in a homogeneous elastic material with sufficient accuracy.

Case 2 is carried out to validate the NMM by simulating a stress wave passing through a rock joint. In order to avoid superposition of the stress waves, the one-dimensional bar is scaled up to 200 m by 5 m. And the incident wave is a half sinusoidal compression wave with a frequency of 50 Hz. A single linear elastic run-through joint is positioned in the center of the elastic bar (at 100 m from the impact end) with a specific stiffness of 5.0 GPa/m. A measuring point B is placed at 50 m from the impact end and another measuring point C is placed adjacent to the right of the joint in order to obtain the waves just before and after the joint (Fig. 3(a)). Other parameters are the same as those in case 1.

Analytical solution of the particle velocity time histories at point B and the joint is derived in order to comparing with the numerical results obtained from the NMM. The particle velocity time history immediately before and after the joint can be derived based on the displacement discontinuity method and the characteristic line theory [8,9,11]. They are given as follows:

(1){\displaystyle \frac{\partial v^{+}(x_J,t-x_J/\alpha )}{\partial t}}={\displaystyle \frac{-\mathrm{2}{v}^{+}(x_J,t-x_J/\alpha )+\mathrm{2}{v}_I(x_B,t-x_B/\alpha )}{z/K_n}},

(2)v^{+}(x_J,t-x_J/\alpha )+v^{-}(x_J,t-x_J/\alpha )=\mathrm{2}{v}_I(x_B,t-x_B/\alpha ),

(3)v_R(x_B,t-(\mathrm{2}{x}_J-x_B)/\alpha )=v^{-}(x_J,t-x_J/\alpha )-v_I(x_J,t-x_J/\alpha ),

where v^{-}(x_J,t-x_J/\alpha ) is the particle velocity time history just before the joint at the coordinate x_J, which is superposed by the incident and reflected waves. v^{+}(x_J,t-x_J/\alpha ) is the particle velocity time history just after the joint, i.e., the transmitted particle velocity. v_I(x_B,t-x_B/\alpha ) is the particle velocity time history of incident wave passing through point B. \alpha, z, K_n represent the wave propagation velocity, mechanical impedance, joint specific stiffness, respectively. v_R(x_B,t-(\mathrm{2}{x}_J-x_B)/\alpha ) is the particle velocity time history of reflected wave at point B.

Given the initial condition v_I(t) and v^{+}(x_J,\mathrm{0}), the particle velocity at the joint position x_J just before and after the joint can be derived numerically based on Eqs. (1) and (2). Figure 3(b) compares the NMM result with the theoretical solution. As seen from Fig. 3(b), the numerical results by using the NMM agree well with the analytical solution not only the peak particle velocities but also the waveforms of the transmitted and reflected waves. The time delay phenomenon caused by the frequency dependent phase lag of the harmonic waves is also simulated. In the time domain, the square of the area enwrapped by stress time history and the time axis indicates the wave energy. When a stress wave travels across a joint, the incident wave energy is divided into two parts, one part of which is reflected and the other is transmitted. The sum of square of the areas enwrapped by transmitted and reflected stress time history equates to the total incident wave energy, which also proves the correctness and precision of the NMM. In fact, in consideration of the reflected wave, the run-through joint acts as a viscoelastic boundary which behaves between a free boundary and an rigid boundary. When the joint stiffness decreases to zero, the joint acts as a free boundary without viscosity, from which the incident energy is completely reflected back, however, when the joint stiffness increases to infinity, the joint acts as a rigid boundary with no elasticity and the incident wave energy is totally absorbed by the downstream material. The results show that the NMM adopted in the present study has satisfactory ability to model the stress wave propagation across jointed rock mass.

As a general form in the natural environment, non-through rock joints with a finite length widely exist in a rock mass. It is also necessary to validate the NMM in simulation of stress wave passing through a non-though joint. A P-wave traveling across a non-through joint in the one-dimensional rock bar is subsequently modeled. It is worth mentioning that the NMM has a special feature that mathematical covers do not necessarily conform to the physical domain boundaries, the validation of the NMM in simulation of stress wave propagation in fractured rock is necessary.

Different from case 2, a non-through joint with finite length as shown in Fig. 4(a) is further simulated. The measuring points B, D and the non-through joint are positioned at 40, 120, and 80 m respectively. And the joint specific stiffness is reduced to 2.5 GPa/m to amplify the attenuation in order to observe the results more obviously. The length of the joint is set to be 0.4, 0.6 and 0.8 times of the width of the slender bar as marked 0.4 H, 0.6 H, and 0.8H (H denotes the width of the rock bar) respectively, and the joint is set in the center of cross section. Figure 4(b) gives the stress waves recorded at point B and point D, which represent the incident wave, reflected wave passing B and transmitted wave passing D. From Fig. 4(b), the incident waves are identical and the transmitted waves are also almost the same, according to which the transmission coefficient is about 99%, while that caused by the run-through joint was 78.6%. The effect of the joint length on the reflected and transmitted waves is minor. Most of the incident wave can pass through the non-through crack by diffractions from the two intact sides of the rock bar. When the stress wave arrives at the joint, the stress distribution is altered due to the stiffness difference between the joint and the intact rock. In order to achieve equilibrium at the cross section, a larger stress is concentrated in the two sides beyond the non-through joint. The larger the joint length, the higher the stress of the two sides is. The average stress of the two intact sides can reach to about 1.6, 1.9 and 2.2 times of the incident peak stress in cases of 0.4H, 0.6 H and 0.8H of the joint lengths, respectively.

Numerical simulation of defected rock

NMM simulation

A rock mass is a highly defected material in a microscopic view, which significantly affects the mechanical behavior, especially the dynamic behavior of the rock mass. The complicated effects of the multiple irregular cracks on the rock mass dynamic properties can hardly be described analytically. The NMM, on the other hand, has the advantage in modeling statistically and heavily crack material using a consistent mathematical cover system.

A half sinusoidal compression wave is applied onto a slender rock bar, which has the same dimension as in the case 1 example. A series of random microcracks are generated in the rock bar to simulate the highly defected rock, which is illustrated in Fig. 5. The whole rock bar is inhomogeneous, in which microcracks are distributed with their lengths varying with a certain fraction of the bar width, and with the uniform distribution of their orientations and center positions. In the NMM, the cracks is modeled by simply splitting a mathematical cover into several physical covers and assigning each physical cover an independent local function, which makes the arbitrarily complex cracks be modeled in a consistent manner. The crack stiffness is described by a contact model whose specific normal and tangential stiffness are modeled by normal and tangential springs, respectively (Fig. 5). Due to the independence of the mathematical covers to the physical domain, the mesh of the NMM is always regular and is not required to coincide with neither the external boundary nor the internal cracks. If singular physical covers are enriched with the asymptotic crack tip functions, crack tips that partially cut the mathematical covers can be modeled [7,12]. In summary, the NMM is suitable to model a highly cracked rock mass in a convenient way.

Different amount of cracks are distributed in the rock bar to simulate the crack quantity effect on the rock dynamic behavior, which is divided into four groups marked Q100, Q200, Q400 and Q600, whose total number of cracks is 100, 200, 400 and 600 respectively. In each crack quantity group, four different crack lengths are simulated; they are 0.2, 0.3, 0.4 and 0.5 times of the width of the rock bar and are marked with 0.2H, 0.3H, 0.4H and 0.5H respectively. Due to the highly discontinuity of the rock bar, the stress caused by the P-wave in the bar is extremely fluctuant and usually is under-represented in a particular area. So, three transverse sections of the bar are selected as measuring sections (0.4, 0.8, and 1.2 m away from the left side respectively) to measure the average stress of the section when the stress wave propagates over it. It should be mentioned that the cracks should not be too close to the measuring section to prevent the stress re-disturbance caused by the cracks.

Viscoelastic behavior and calculation of storage modulus and loss modulus

According to the one-dimensional harmonic wave propagation theory, the axial motion equation and the constitutive equation of a stress wave propagating across a straight, slender bar with density \rho and dynamic complex modulus E^{*}(\omega ) can be expressed as

(4){\displaystyle \frac{{\partial }^{\mathrm{2}}\widetilde{ϵ}(x,\omega )}{\partial x^{\mathrm{2}}}}+{\displaystyle \frac{\rho {\omega }^{\mathrm{2}}}{E^{*}(\omega )}}\widetilde{ϵ}(x,\omega )=\mathrm{0},

(5)\tilde{\sigma }(x,\omega )=E^{*}(\omega )·\widetilde{ϵ}(x,\omega ),

where \omega is the angular frequency of the component harmonic wave that can be obtained from wave frequency f by \omega =\mathrm{2}\mathrm{\pi }·f. \widetilde{ϵ}(x,\omega ) and \tilde{\sigma }(x,\omega ) are strain and stress variables in the frequency domain. The rock dynamic complex modulus E^{*}(\omega ) can be expressed as E^{*}(\omega )=E^{\prime }(\omega )+i{E}^{″}(\omega ), in which i denotes the imaginary sign. And the real part E^{\prime }(\omega ) of the dynamic complex modulus E^{*}(\omega ) is the dynamic storage modulus which measures the stored energy, representing the elastic portion. While the imaginary part E^{″}(\omega ) measures the energy dissipated, representing the viscous portion.

In order to calculate the storage modulus and loss modulus of the microcracked rock mass, the propagation coefficient should be introduced, which can be defined as \gamma (\omega )=\alpha (\omega )+ik(\omega ), where the coefficient \alpha (\omega ) is the attenuation coefficient, and k(\omega ) is the wave number. And the propagation coefficient \gamma (\omega ) and complex dynamic modulus have the relation as E^{*}(\omega )=-\rho {\omega }^{\mathrm{2}}/{\gamma }^{\mathrm{2}}(\omega ). Comparing the real and imaginary parts of the above equation, the storage modulus and loss modulus can be obtained as follows,

(6)E^{\prime }(\omega )={\displaystyle \frac{\rho {\omega }^{\mathrm{2}}(k^{\mathrm{2}}-{\alpha }^{\mathrm{2}})}{{(k^{\mathrm{2}}+{\alpha }^{\mathrm{2}})}^{\mathrm{2}}}},

(7)E^{″}(\omega )={\displaystyle \frac{\mathrm{2}\rho {\omega }^{\mathrm{2}}k\alpha }{{(k^{\mathrm{2}}+{\alpha }^{\mathrm{2}})}^{\mathrm{2}}}}.

When the incident wave and the attenuated wave are available, the attenuation coefficient \alpha (\omega ) and the wave number k(\omega ) are obtained from the following equations [13]:

(8)\alpha (\omega )=\mathrm{Re}\left[\ln \left({\displaystyle \frac{\mathfrak{F}(\left|{ϵ}_{\mathrm{2}}\right|)}{\mathfrak{F}(\left|{ϵ}_{\mathrm{1}}\right|)}}\right)/l\right],

(9)k(\omega )=\mathrm{Im}\left[\ln \left({\displaystyle \frac{\mathfrak{F}(\left|{ϵ}_{\mathrm{2}}\right|)}{\mathfrak{F}(\left|{ϵ}_{\mathrm{1}}\right|)}}\right)/l\right],

where \mathfrak{F} denotes the treatment of applying the Fourier transformation, Re and Im denote real and imaginary parts of the complex expression, respectively. {ϵ}_{\mathrm{1}} and {ϵ}_{\mathrm{2}} denote the incident strain pulse and the attenuated pulse, and l denotes the wave traveling distance between these pulses.

Results and discussions

Figure 6 shows a typical result from the numerical simulation, from which obvious stress wave attenuation as well as time delay could be observed. This phenomenon reveal that the heavily microcracked rock mass has viscoelasticity to the stress wave due to the effect of multiple reflection, diffraction and scattering on the incident wave when there exist intrinsic microcracks in the rock mass. The viscosity that is caused by these microcracks will dissipate the wave energy and slow down the wave velocity as well.

From a serial numerical results, the storage modulus and loss modulus which are the main indicators of the viscoelasticity of the microcracked rock mass can be calculated, and their relations with microcracks and incident wave frequency are shown in Figs. 7-10. From these figures, the common trend can be revealed that the storage modulus increases sharply with the increasing of angular frequency at the initial stage when the angular frequency is relatively small and arrives at a peak value then keeps stable when the frequency keeps increasing. While the loss modulus also rockets to a peak value at the first stage, however, it plunges after it reaching its peak value at the second stage, then goes to a stable value at the third stage when the frequency increases further. Both the storage modulus and loss modulus are frequency dependent.

Figure 7 shows that the dynamical complex modulus of the microcracked rock mass can be affected by the microcrack length in the rock mass significantly. The storage modulus will reach a much larger stable value as well as a much steeper slope at the initial stage with the decreasing of the crack length. While the shorter the cracks are, the higher peak value the loss modulus reaches at its first stage and lower stable value it holds at the stable stage. And the decay of the loss modulus is much faster when the crack length is relatively short. These indicate that the increase of the crack length will degrade the overall elasticity of the microcracked rock mass but enhances the viscosity, which will dissipate much more wave energy and cause much time delay when wave propagates. In fact, the wave time delay relates to the phase lag \theta (\omega ) of the component harmonic waves which can be calculated from tan \theta (\omega )=E^{″}(\omega )/E^{\prime }(\omega ). Theoretically, when the crack length approaches to zero, the viscosity in the rock will decrease gradually and the storage modulus will approach to the Young’s modulus E, while the loss modulus at the stable stage will approach to zero.

The relation between dynamic complex modulus and the quantity of the cracks as presented in Fig. 8 shows the similar tendency as the effects of the length of the cracks on the dynamic complex modulus. The increase of the quantity of the microcracks will increase the loss modulus and decrease the storage modulus. When the crack quantity is 100, the storage modulus reaches to about 68 GPa, which is a little smaller than the intact rock Young’s modulus (75 GPa). In comparison, when the crack quantity rises to 600, the storage modulus can only approach about 28 GPa. The overall elasticity can be greatly degraded by the presence of the microcracks. The loss modulus also increase from about 1 GPa to about 2.5 GPa as the quantity increases from 100 to 600, which shows that although the increase of the absolute value of E^{″} is smaller than the decrease of the absolute value of E^{\prime }, the changing ratios of them are similar.

Figure 9 shows influence of variation of crack specific stiffness on the viscoelasticity of the microcracked rock. When the specific stiffness of the cracks increases, less energy will be reflected. Therefore, the increase of the specific stiffness rehabilitates the elasticity of the microcracked rock mass. The changing of storage modulus as well as loss modulus is within a relative small range.

Figure 10 illustrates that the material viscoelasticity of the microcracked rock mass is independent of the incident wave frequency. Although much wave attenuation is observed as the incident wave frequency increases, the viscoelasticity does not vary according to the incident wave frequency. The dynamic complex modulus only changes with the component harmonic wave frequency. When the incident wave frequency is relatively low, most parts of component harmonic wave are mainly in the low frequency range. While the incident wave frequency is increased, the component harmonic wave of the incident wave extends to a broader range. According to the theory of wave propagation across cracks, the transmission coefficient will decrease as the harmonic wave frequency increases. Therefore, much energy will be dissipated when the stress wave propagates across the microcracked rock mass if the incident wave frequency increases. However, the dynamic modulus does not alter with the incident wave frequency. It is the intrinsic property of the heavily microcracked rock mass.

Conclusions

NMM is suitable in solving static and dynamic problems in both continuous and discontinuous materials. Its description of discontinuity is convenient due to the conception of dual cover systems (mathematical and physical covers). And the validated NMM has shown sufficient accuracy in the simulation of stress wave propagation.

The simulation of P-wave propagation across a non-through crack is conducted to investigate the transmission, reflection and diffraction of a stress wave. The diffraction phenomenon is the main contribution to the transmission coefficient as the stress wave passing the non-through joint. The transmission coefficient of the wave propagation through a non-through joint is much larger than that of a run-through crack.

In addition, the present study uses NMM to investigate the dynamic stress wave behavior when P-wave transmits highly microcracked rock mass. It shows that the highly microcracked rock mass has remarkable viscoelastic property to the stress wave propagation, which can be verified by the attenuation of the wave amplitude and wave time delay when it propagates through the defected rock mass with multiple uniformly distributed microcracks.

The dynamic complex modulus which indicates the viscoelasticity is significantly affected by the crack length, quantity, and stiffness. However, it is independent of the incident wave frequency. When either the length or quantity of the micro cracks increases, they remarkably degrades the overall elasticity of the microcracked rock mass; meanwhile, enhances its viscosity. That is because the increasing discontinuity will make the diffraction much more difficult. Thus, when the length and the quantity of cracks increase, the viscosity will be intensified due to the multiple reflection and dispersion. The decrease of the crack stiffness also intensifies the discontinuity of the rock mass, which makes viscosity more significant. Nevertheless the incident wave frequency has no influence on viscosity of the microcracked rock mass, which is the intrinsic property of a material, although the incident wave could be attenuated more greatly when the incident wave frequency increases. In conclusion, the microcracks in the elastic rock mass will cause viscoelasticity of the overall microcracked rock mass, which would significantly attenuate the stress wave and make a travel time delay when the stress wave propagates.

References

[1] Ichikawa Y, Kawamura K, Uesugi K, . Micro- and macrobehavior of granitic rock: observations and viscoelastic homogenization analysis. Computer Methods in Applied Mechanics and Engineering, 2001, 191(1-2): 47-72

[2] Pyrak-Notle L J. Seismic visibility of fractures. Dissertation for the Doctoral Degree. Berkeley: University of California, 1988

[3] Cook N G W. Natural Joints in Rock- Mechanical, Hydraulic and Seismic Behavior and Properties under Normal Stress. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 1992, 29(3): 198-223

[4] Li J, Ma G, Zhao J. An equivalent viscoelastic model for rock mass with parallel joints. Journal of Geophysical Research, 2010, 115(B3): B03305

[5] Mal A K. Interaction of elastic waves with a Griffith crack. International Journal of Engineering Science, 1970, 8(9): 763-776

[6] Shi G H. Manifold method of material analysis. In: Proceedings of Transactions of the 9th Army Conference on Applied Mathematics and Computing. Minneapolis, 1991: 57-76

[7] Ma G W, An X M, He L. The numerical manifold method: a review. International Journal of Computational Methods, 2010, 7(1): 1-32

[8] Miller R K. Approximate method of analysis of transmission of elastic-waves through a frictional boundary. Journal of Applied Mechanics-Transactions of the ASME, 1977, 44(4): 652-656

[9] Schoenberg M. Elastic wave behavior across linear slip interfaces. Journal of the Acoustical Society of America, 1980, 68(5): 1516-1521

[10] Pyraknolte L J, Myer L R, Cook N G W. Transmission of seismic-waves across single natural fractures. Journal of Geophysical Research-Solid Earth and Planets, 1990, 95(B6): 8617-8638

[11] Zhao J, Cai J G. Transmission of elastic P-waves across single fractures with a nonlinear normal deformational behavior. Rock Mechanics and Rock Engineering, 2001, 34(1): 3-22

[12] An X. Extended numerical manifold method for engineering failure analysis. Dissertation for the Doctoral Degree. Singapore: Nanyang Technological University, 2010

[13] Bacon C. An experimental method for considering dispersion and attenuation in a viscoelastic Hopkinson bar. Experimental Mechanics, 1998, 38(4): 242-249

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