197479775
Transmission of Normal P-Waves across a Single Joint Based ong‐λModel
This paper investigates the wave transmission and reflection of an elastic P-wave at a single joint for normal incidence. First, considering a coupled joint (correction parameterλ,0<λ<1), a normal deformation constitutive model of the joint (g‐λmodel) under static or quasi-static loading is introduced and then extended to dynamic loading. The nonlinearity of the joint stress-deformation curve increases with increasingλ. Second, the interaction between the P-wave and the joint is investigated by using the method of characteristics and the displacement discontinuity method to deduce the differential expression of the transmitted wave’s particle velocity. The approximate analytical expressions of the transmission and reflection coefficients are obtained according to the Lemaitre equivalent strain assumption. Third, parametric studies are conducted to evaluate the effects ofλon transmission characteristics for a normally incident P-wave at a single joint. The results show that the particle velocity of the transmitted wave depends onλ. Whenλtakes the limit values 0 and 1, the transmitted wave’s particle velocities are then consistent with the conclusions of the classical exponential model and the Barton–Bandis model. In addition, the transmission and reflection coefficients are discussed with respect toλand also to the ratio of the joint closure to the maximum allowable joint closure.
129313178
Effects of weak layers on particle velocity measurements
SummaryResults are presented from a testing program to study the effect weak layers embedded in a strong rock strata have on particle velocity when subjected to explosive loading. A similar computational study had been conducted earlier with WONDY — a finite difference Langrangian code developed at Sandia National Laboratory. The experiments were conducted using models fabricated from Hydrocal containing a single dry sand layer or clay layer through which the stress wave traveled. Particle velocity was measured in front of and past the weak layer to determine attenuation, pulse shape changes, and displacement loss. The results from the model testing indicated that particle velocity amplitude decreased significantly when the stress wave passed through the weak layer. The velocity pulse width on the other hand was found to remain relatively constant when passing through the weak layer. The computational results from WONDY predicted similar behavior and hence were in good agreement with the tests. In the experiments, the velocity loss across a sand layer was found to be much larger than the loss across a clay layer. The stress wave velocity in the sand layer was found to be significantly smaller than in the Hydrocal while the experimentally determined wave velocity in the clay was nearly equal to the wave velocity in the Hydrocal.
121672131
On modelling of incident boundary for wave propagation in jointed rock masses using discrete element method
Abstract Numerical modelling is an effective alternative for studying the mechanical behaviour of rock masses. Very often, in solving the same problem with the same code, different models yield remarkably different results. There are many reasons. Amongst them, the boundary conditions are often overlooked but it has significant influence on the stress wave propagation in jointed rock masses. Usually, in solving dynamic problems, the incident boundaries are described either by stress-history input (SHI) or velocity-history input (VHI). This paper presents the study on the effect of these two approaches using discrete element method (DEM) on wave propagation and wave attenuation in jointed rock masses. Two cases were designed to illustrate the points. The first one shows one-dimensional P-wave propagation along a rock bar having an internal cross joint. The second one shows the blasting wave propagating through jointed rock masses. The blasting wave is induced by an underground explosion. The rock bar/masses are discretized into elements and the analyses adopt the DEM, which is available in the code ‘UDEC’. The results reveal that the VHI approach can yield results in accord with field-test data, while the SHI approach may lead to unreasonable results. The two approaches may yield the same results for homogeneous rock, but definitely not the same for jointed rock masses. In addition to the numerical illustrations, this paper also presents some insights of the reason behind it.
120836830
The numerical manifold method for elastic wave propagation in rock with time-dependent absorbing boundary conditions
Abstract In this study, a modified first-order Higdon׳s absorbing boundary scheme is proposed and incorporated into the numerical manifold method (NMM) to reduce reflections from artificial boundaries induced by truncating infinite media. The modified time-dependent absorbing boundary scheme can not only consider the absorbing boundary and input boundary at the same artificial boundary, but also take the effects of the incident angles into consideration by adjusting the velocities and strains of points at the boundary automatically. For illustrating the efficiency of the proposed time-dependent absorbing boundary scheme, comparisons between the results of the proposed method and the widely used viscous boundary conditions for different incident angles are presented. The developed NMM is then used to investigate wave attenuation and transmission across a joint in an infinitely long rock bar.
123956378
Experimental and Numerical Investigations on Feasibility and Validity of Prismatic Rock Specimen in SHPB
The paper presents experimental and numerical studies on the feasibility and validity of using prismatic rock specimens in split Hopkinson pressure bar (SHPB) test. Firstly, the experimental tests are conducted to evaluate the stress and strain uniformity in the prismatic specimens during impact loading. The stress analysis at the ends of the specimen shows that stress equilibrium can be achieved after about three wave reflections in the specimen, and the balance can be well maintained for a certain time after peak stress. The strain analysis reveals that the prismatic specimen deforms uniformly during the dynamic loading period. Secondly, numerical simulation is performed to further verify the stress and strain uniformity in the prismatic specimen in SHPB test. It indicates that the stress equilibrium can be achieved in prismatic specimen despite a certain degree of stress concentration at the corners. The comparative experiments demonstrate that the change of specimen shape has no significant effect on dynamic responses and failure patterns of the specimen. Finally, a dynamic crack propagation test is presented to show the application of the present work in studying fracturing mechanisms under dynamic loading.
123059709
P-wave transmission across fractures with nonlinear deformational behaviour
Note: Sols Reference LMR-ARTICLE-2006-015doi:10.1002/nag.515View record in Web of Science Record created on 2006-11-09, modified on 2016-08-08
129066631
Study of wave attenuation across parallel fractures using propagator matrix method
Stress wave attenuation across fractured rock masses is a great concern of underground structure safety. This paper presents an analytical study on wave attenuation across parallel fractures at arbitrary incidence angles, where multiple reflections occurring between fractures are taken into account. Combined with displacement discontinuous model, plane wave analysis and propagator matrix method are applied to develop relations between the first layer and the nth layer with respect to potential amplitudes or displacements and stresses in matrix form. With initial and boundary conditions for different scenarios, potential amplitudes in any layer or displacements and stresses at any point can be obtained by solving corresponding matrixes. After parametric studies, it is found that parameters including incidence angle, normalized fracture stiffness, number of fractures, and fracture spacing have obvious effects on wave attenuation across parallel fractures.
129832929
Some Fundamental Issues and Verification of 3DEC in Modeling Wave Propagation in Jointed Rock Masses
Discontinuities in rock masses greatly influence stress wave transmission. Before performing numerical modeling on complicated cases, benchmarking modeling must be performed first to validate the numerical code through comparison with theoretical solutions or experiment results. The accuracy of numerical results generally increases with decreasing mesh size. However, with an extremely fine mesh size, substantial computational time will be consumed. 3D plane wave propagation across a single joint or a joint set can be treated equivalently to a corresponding 2D case, as shown both analytically and numerically. However, it should be noted that when the incident wave is not a plane wave or there are unparallel joints and joint sets, a 3D case cannot be equivalent to a 2D case.
31905801
A further study on seismic response of a set of parallel rock fractures filled with viscoelastic materials
The purpose of this study is to further investigate the seismic response of a set of parallel rock fractures filled with viscoelastic materials, following the work by Zhu et al. Dry quartz sands are used to represent the viscoelastic materials. The split Hopkinson rock bar (SHRB) technique is modified to simulate 1-D P-wave propagation across the sand-filled parallel fractures. At first, the displacement and stress discontinuity model (DSDM) describes the seismic response of a sand-filled single fracture. The modified recursive method (MRM) then predicts the seismic response of the sand-filled parallel fractures. The SHRB tests verify the theoretical predictions by DSDM for the sand-filled single fracture and by MRM for the sand-filled parallel fractures. The filling sands cause stress discontinuity across the fractures and promote displacement discontinuity. The wave transmission coefficient for the sand-filled parallel fractures depends on wave superposition between the fractures, which is similar to the effect of fracture spacing on the wave transmission coefficient for the non-filled parallel fractures.
129543901
Effects of a Single Open Joint on Energy Transmission Coefficients of Stress Waves with Different Waveforms
Open joints have significantly different effects on stress wave propagation across them as compared to closed and filled joints. In the present study, a theoretical model is developed based on the analysis of interaction process between stress waves and a single open joint. The analytical solutions to the energy transmission coefficient are mathematically derived for stress waves across an open joint. Parametric studies are conducted to evaluate the effects of various parameters on the energy transmission coefficient. It is found that the energy transmission coefficient follows a similar trend for all types of stress waves, but stress waves with different waveforms have different values. The energy transmission coefficient increases with the increase in the wave amplitude and duration, but decreases with the gap width. It also increases at the outset, and then decreases gradually with the increase in the incident angle for rectangle, symmetric triangle and ascending triangle waves, but it decreases with the incident angle for sine and descending triangle waves. Furthermore, different-shaped stress waves have different critical gap widths for wave transmission. The optimal incident angle and the critical gap width increase as the wave duration increases, but the critical stress amplitude decreases.
136333727
An SHPB test study on wave propagation across rock masses with different contact area ratios of joint
Abstract The Split Hopkinson Pressure Bar (SHPB) apparatus was adopted to investigate the influence of joint contact area and spatial geometry of joint surface on the dynamic property of rock joint and wave propagation. The specimen was comprised of two rock cylinders. The top surfaces of the two cylinders contacted with each other, and their bottom surfaces contacted with the input and output bars of the SHPB apparatus, respectively. One top surface of one cylinder was sawn to shape a number of notches, while the other surfaces of the two cylinders were smooth and flat. The artificial rock joint was modeled as the contacted top surfaces of the two cylinders. The area ratio of contact between joint surfaces equals to the joint matching coefficient (JMC). The incident, transmitted and reflected waves were recorded from the strain gauges mounted on the input and output bars. Then, the transmission and reflection coefficients for strain wave propagation across the rock specimen were obtained. Based on the basic theory of SHPB tests, the stress on the specimen, the deformation of rock specimen and the stress-closure relation of joint were analyzed. The experimental results show that the JMC and the spatial geometry of joint surface affect not only the dynamic behavior of joint but also the stress wave propagation.
134868771
A rapid and nondestructive method to determine normal and shear stiffness of a single rock joint based on 1D wave-propagation theory
ABSTRACTDetermination of joint stiffness, one of the most important mechanical properties of rock joints, is of great significance. However, joint stiffness is often difficult to complete and accurate determine because of the 3D nature of joints and limited budgets and visible exposure. Wave attenuation in a rock mass is mainly attributed to the presence of joints. Based on 1D plane-wave theory, a rapid and nondestructive method, namely, the rapid evaluation method, is proposed to calculate normal and shear stiffness of joints. The joint is viewed as a boundary condition modeled by the displacement discontinuity model. By solving wave equations with reasonable approximation, joint stiffness can be inversely solved, which is dependent on the seismic impedance, transmission or reflection coefficient, and dominant frequency of transmitted waves. Ultrasonic laboratory tests were carried out to record incident, reflected, and transmitted waveforms as input to the rapid evaluation method. It was found that the ...
135178297
An SHPB Test Study on Stress Wave Energy Attenuation in Jointed Rock Masses
A series of dynamic tests were carried out to investigate the effect of joint roughness on the wave energy attenuation in rock masses and estimate the relation between joint roughness and seismic quality factor of rock masses. The modified split Hopkinson pressure bar (SHPB) apparatus was adopted in this study, where the loading, input and output bars were made of gypsum. The propagation coefficient of the gypsum bars was measured from trial tests. According to the propagation coefficient of the gypsum bars, the strain, stress and particle velocity on the contact surfaces between the specimen and input/output bars were obtained from the test data recorded by the strain gauges. The specimens were prepared by a three-dimensional printer with plaster and binder. Each specimen modeled a rock mass with one joint with different roughness. The seismic quality factor of specimen is also estimated from the proposed approach of wave energy dissipation. The effects of joint roughness on the seismic quality factor of rock mass and the wave energy attenuation across the rock mass are analyzed from test results.
117181091
Experimental investigation of thermal effects on dynamic behavior of granite
Abstract A systemic understanding of the thermal effects on the dynamic behavior of granite is significant to thermal engineering applications such as waste disposal engineering and underground coal gasification. In the present study, scanning electron microscope (SEM) tests were carried out to evaluate the thermal effects on the geophysical properties of granite. The results show that the density decreases slightly as the temperature increases from 25 °C to 400 °C but that it decreases sharply as the temperature increases further to our maximum tested temperature of 800 °C. The defect rate increases slightly as temperature increases from 25 °C to 400 °C and then increases sharply as the temperature further increases to 800 °C. Next, ultrasonic wave tests were performed to evaluate the thermal effects on the wave velocity and P-wave modulus. The results show that both the wave velocity and P-wave modulus decrease sharply and linearly below the temperature of 400 °C, before deceasing nonlinearly as the temperature increases to 800 °C. Finally, split Hopkinson pressure bar (SHPB) tests were adopted to investigate the thermal and loading rate coupling effects. The results show that the dynamic strength decreases linearly as temperature increases but increases as the impact pressure increases. However, the dynamic energy absorption capacity increases below 400 °C but then decreases as the temperature increases to 800 °C. The thermal effects on energy absorption capacity are more obvious for granite under a smaller impact pressure.
133990180
Wave transmission across linearly jointed complex rock masses
Abstract This paper presents an investigation of wave transmission in complex rock masses; the rocks have different wave impedances on either side of a joint. The conventional displacement discontinuity method was modified for complex rock mass wave propagation by introducing four sets of characteristic lines. Two cases, “soft-to-hard” and “hard-to-soft” rock masses, were used to demonstrate wave propagation through the complex rock masses. The results show that the wave impedance ratio plays an important role in the wave propagation through the complex rock masses. For the case of “soft-to-hard” rock masses, the particle velocity transmission ratios are always smaller than 1.0, while the stress transmission ratios may be larger than 1.0 when the wave impedance ratio and the frequency of the incident wave are sufficiently small. On the other hand, for the case of “hard-to-soft” rock masses, the stress transmission ratios are always smaller than 1.0, while the particle velocity transmission ratios may be larger than 1.0 when the wave impedance ratios are sufficiently large and the frequencies of the incident wave are sufficiently small. The phenomenon indicates that wave propagation in complex rock masses is not always attenuated but may actually be enhanced in some cases.
140599633
Methods of Geological Engineering in Discontinuous Rocks
Keywords: Mecanique des roches ; Methode des elements finis Reference Record created on 2004-09-07, modified on 2016-08-08
140672221
FUNDAMENTALS OF ROCK JOINT DEFORMATION
Abstract This paper describes laboratory investigations of the deformation characteristics of rock joints under normal and shear loading. Normal deformability was studied by conducting loading/unloading and repeated load cycling tests on a wide variety of fresh and weathered joints in five different rock types. The data invariably showed non-linear behaviour, irrespective of the rock and joint type. A hyperbolic function is suggested to describe the stress-closure/opening curves of joints. Quantitative relations between normal deformability and relevant joint parameters (aperture, wall strength and roughness) are developed. Tentative conclusions on the changes in normal stiffness during shearing are also presented. The behaviour of dislocated (mismatching) joints is studied qualitatively and analytically. Shear deformability was studied by performing direct shear tests under normal stresses in the range of engineering interest. It is shown that behaviout in the pre-peak range is invariably non-linear depending on the joint type, and can be adequately described by easily measured parameters and hyperbolic functions.
129864427
Transmission of Elastic P-waves across Single Fractures with a Nonlinear Normal Deformational Behavior
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 effects 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 (|Tnon|) and reflection (|Rnon|) coefficients, 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 effects of the nonlinear fracture normal deformation on P-wave transmission, in terms of initial normal stiffness 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, |Tlin| and |Rlin| for the linear model are special solutions of |Tnon| and |Rnon| 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 |Tnon| from |Tlin|.
123438287
Effects of multiple parallel fractures on apparent attenuation of stress waves in rock masses
Abstract Stress waves propagating through multiple parallel fractures are attenuated (and slowed) due to multiple wave reflections and transmissions at the fractures. This paper presents a theoretical study and the UDEC modeling on the effects of multiple parallel planar fractures on the apparent attenuation of normally incident one-dimensional elastic waves. The case of normal incidence of waves is studied, because we want to remove the influence of the incident angle and to focus purely on the effects of stiffness, spacing and number of parallel fractures. In the theoretical study, an approach is developed to explicitly take into account the attenuative effect of each fracture with the displacement discontinuity model, and to implicitly consider complex interfracture multiple wave reflections with the method of characteristics. The intrinsic attenuation mechanisms are neglected so that the attenuative effects of the multiple reflections can be concentrated on. This approach does not lose the discreteness of wave attenuation at individual fractures, and avoids the difficulty in explicitly determining the complex process of superposition of multiple reflected and transmitted wave fields. With this approach, a set of recurrence equations with respect to particle velocities before and after the fractures are established. These equations are numerically solved with sufficient accuracy. In analysis, there is no restriction to large fracture spacing or to small fracture spacing compared with wavelength. The magnitude of transmission coefficient (∣ T N ∣) for waves transmitting normally across a set of multiple parallel fractures is calculated as a function of the ratio ( ξ ) of fracture spacing to wavelength, for different normalized stiffness ( k / zω ) and for a different number of fractures. Parametric studies are conducted to examine the effects of multiple parallel fractures on wave attenuation, especially in terms of the spacing and the number of fractures. It is shown that the dependence of ∣ T N ∣ on the fracture spacing and the fracture number is governed by ξ . In addition, the effects of multiple wave reflections on ∣ T N ∣ are quantitatively discussed for different values of ξ . In the numerical modeling, the same problem is studied with a discontinuum-based numerical method, termed as the Discrete Element Method (DEM). The Universal Distinct Element Code (UDEC) is used to model one-dimensional wave propagation in rock masses containing no fractures, a single planar fracture and multiple parallel planar fractures. The modeling results of the transmission coefficient are compared with the theoretical solutions. An agreement between them has been achieved. It is verified that the UDEC is capable of modeling one-dimensional wave propagation across multiple parallel fractures.
129552859
Stress wave transmission across a filled joint with different loading/unloading behavior
Abstract This paper presents a theoretical study of stress wave propagation across a rock joint with different loading/unloading behavior. Two Bandis–Barton (B–B) models are introduced to describe the stress–closure relationships of a rock joint under loading and unloading situations, respectively. A characteristics method is proposed to investigate the stress wave transmission. Amplitude-dependence and frequency-dependence of unloading effect on the energy transmission are analyzed. The unloading effect on stress waves with different waveforms and stress waves with multiple pluses are evaluated. The results show that the characteristics method can be used to investigate the wave propagation across a rock joint with different loading/unloading behavior. The effect of unloading behavior on energy transmission is amplitude- and frequency-dependent. The stress wave with a rectangular waveform is less influenced by the unloading behavior than the stress waves with other waveforms. Unloading behavior has a significant effect on the energy transmission for the stress wave with both single pulse and multiple pulses. However, unloading behavior has an effect on the wave attenuation provided that the time interval of multiple pulses stress wave is sufficiently small.
126900289
Seismic wave propagation through an in-situ stressed rock mass
Abstract This paper presents an analytical study of seismic wave propagation through an in-situ stressed rock mass. The joint deformation is assumed to satisfy the nonlinear Bandis–Barton (B–B) deformational model. The displacement discontinuity method (DDM) is introduced to take account of the effect of in-situ stress. The amplitude- and frequency-dependence of seismic wave propagation through an in-situ stressed rock mass are analyzed. The effects of in-situ stress on seismic attenuation, energy transmission and effective seismic velocity are discussed. The results show that seismic attenuation, energy transmission and effective seismic velocity are influenced by seismic wave amplitude and frequency, and in-situ stress. Moreover, the amplitude- and frequency-dependence of a seismic wave obtained by conventional DDM are special cases when in-situ stress is not considered. The effect of in-situ stress on seismic attenuation, energy transmission and effective seismic velocity can be neglected when either the amplitude or the frequency is sufficiently large. When the frequency is sufficiently small, the effect of in-situ stress on seismic attenuation and energy transmission can also be neglected, but its effect on the effective seismic velocity cannot be ignored.
129314406
ROCK JOINT COMPLIANCE TESTS FOR COMPRESSION AND SHEAR LOADS
Abstract A series of compression and shear tests were conducted at large scale on joint surfaces in granite and slate specimens, with special equipment being used to isolate the deformations of the joints from those of the rock material. The matrix of compliance components for a rock joint is discussed in relation to these standard tests and it is shown that one pair of compliance components can be determined from a compression test and a second pair from a shear test. Any attempt to base predictions of rock mass behaviour on the combined use of these two pairs of components should take into account the different stress paths to which they relate. The nature of the contacts between the asperities on adjacent joint surfaces is considered as a means of suggesting the form of the stress-deformation relations, both in compression and shear. This approach is shown to be of great assistance in helping to explain the test results and fit the observed compliance characteristics. For the compression tests, the test results indicate a pattern of decreasing compliance with load, an increase in compliance with initial apeture, and high levels of recoverability of deformation. In the shear tests, the curves for the relative displacements indicate there are three zones of different behaviour, elastic, transition and sliding. There appear to be links between the way in which forces are transferred at asperities and the typical patterns of observed compliance.
120870391
Time-dependent closure of a fracture with rough surfaces under constant normal stress
Abstract Time-dependent closure of a fracture with rough surfaces subjected to stepwise normal stress was considered theoretically by viscoelastic modeling of rock. A formula for the relationship between constant normal stress and time-dependent closure as a function of time was derived based on the aperture distributions of a fracture and the relaxation modulus YE′(t) of rock. Theoretical consideration showed that the ultimate closure of a fracture under constant normal stress can be estimated from the normal stress–elastic closure curve by using the values of the relaxation modulus at t=0 and ∞, and that the ultimate time-dependent closure is independent of the normal stress if the elastic closure is linear with the logarithm of the normal stress. Experiments and a Monte Carlo simulation on time-dependent closure under constant normal stress were conducted for a hydraulic fracture created in granite in the laboratory to provide the verification of the theory. The results obtained in the experiments showed that the ultimate time-dependent closure of a hydraulic fracture was almost independent of the normal stress when the elastic closure was linear with the logarithm of the normal stress. A Monte Carlo simulation on time-dependent closure of a fracture under constant normal stress showed that time-dependent closure of a fracture for which the elastic closure is linear with the logarithm of the normal stress does not depend on the normal stress because the increase in contact area during time-dependent closure increases with the normal stress.
121419191
Models for normal fracture deformation under compressive loading
Abstract A new semi-empirical model that can be used to predict fracture deformation behavior under normal compressive loading is presented. The development of a simple exponential model is presented first after which a modified and more general exponential model, with an additional degree of freedom in the model parameters, is obtained. The simple and the modified exponential models are then compared to available fracture closure models, namely the empirical Barton–Bandis hyperbolic model, and a power-law model based on Hertzian contact theory, to determine how good they fit the results of fracture closure experiments under monotonically increasing normal compressive loading. A new parameter called the half-closure stress, σ 1/2 , is introduced and is used, in addition to the maximum fracture closure, Δ v m , in the model fitting procedures for the Barton–Bandis and the simple exponential model. The half-closure stress is shown to be related to the initial normal stiffness, K ni , used in the original Barton–Bandis model. An additional parameter, n, is used in fitting the modified exponential model to the experimental data. Of the models presented herein, the modified exponential model was found to provide the best fit to the experimental data, for the same values of σ 1/2 and Δ v m , over the entire range of compressive stresses. The power-law model based on Hertzian contact theory was found to be unsuitable for predicting normal fracture deformation behavior.
125020999
Improved nonlinear elastic constitutive model for normal deformation of rock fractures
Some nonlinear elastic constitutive models for normal deformation of rock fracture were reviewed and the significant deviation of these models from experimental results at median stress level was pointed out.A new concept named half-closure stress σ1/2 was introduced to lay emphasis on the analysis of the mathematical defects of both the BB model and the classical exponential model.The generalized exponential model proposed by Malama and Kulatilake(2003) was treated as an example for further explanation of the necessity for model improvement,meanwhile the deficiency of physical meaning of the initial normal stiffness of fracture Kni because of the modification by accession of power function n on σn/σ1/2 was analyzed.Based on the above researches,a governing differential stress-displacement equation of fractures under monotonically raising normal compressive loading was established,and the concept of maximum allowable closure of pseudo fracture Dmax=ξdmax was defined.Then a new 3-parameter constitutive model which could improve the nonlinear elastic normal deformation behavior of fractures was proposed.It was strictly proved that the BB model and the classical exponential model were two special cases of the new model.The new parameter was easy to be determined,and the mathematical defects of the BB model and the classical exponential model were overcome.Counteracting the deviation of the simulated results of these traditional nonlinear models with the test results at median stress level,the physical meaning of Kni would not be lost with the accession of the new parameter.The predicated results fitted with the test results well in adopting the new model for the predication of other test data,which verified the feasibility of the new model.
121527648
A new constitutive law for the nonlinear normal deformation of rock joints under normal load
In view of the deviation of the fitting results of the classical exponential model and the hyperbolic model (the BB model) from several experiment data during intermediate stress period, a new constitutive model for the nonlinear normal deformation of rock joints under normal monotonous load is established with flexibility-deformation method. First of all, basic laws of the deformation of joints under normal monotonous load are discussed, based on which three basic conditions which the complete constitutive equation for rock joints under normal load should meet are put forward. The analysis of the modified normal constitutive model on stress-deformation curve shows that the general exponential model and the improved hyperbolic model are not complete in math theory. Flexibility-deformation monotone decreasing curve lying between flexibility-deformation curve of the classical exponential model and the BB model is chosen, which meets basic conditions of normal deformation mentioned before, then a new normal deformation constitutive model of rock joints containing three parameters is established. Two main forms of flexibility-deformation curve are analyzed and specific math formulas of the two forms are deduced. Then the range of the parameters in the g-δ model and the g-λ model and the correlative influence factor in geology are preliminarily discussed. Referring to different experiment data, the validating analysis of the g-δ model and the g-λ model shows that the g-λ model can be applied to both the mated joints and unmated joints. Besides, experiment data can be better fit with the g-λ model with respect to the BB model, the classical exponential model and the logarithm model.
16994513
A thin-layer interface model for wave propagation through filled rock joints
The present study essentially employs a thin-layer interface model for filled rock joints to analyze wave propagation across the jointed rock masses. The thin-layer interface model treats the rough-surfaced joint and the filling material as a continuum medium with a finite thickness. The filling medium is sandwiched between the adjacent rock materials. By back analysis, the relation between the normal stress and the closure of the filled joint are derived, where the effect of joint deformation process on the wave propagation through the joint is analyzed. Analytical solutions and laboratory tests are compared to evaluate the validity of the thin-layer interface model for filled rock joints with linear and nonlinear mechanical properties. The advantages and the disadvantages of the present approach are also discussed.
55019320
Elastic wave behavior across linear slip interfaces
A model for an imperfectly bonded interface between two elastic media is proposed. Displacement across this surface is not required to be continuous. The displacement discontinuity, or slip, is taken to be linearly related to the stress traction which is continuous across the interface. For isotropic interface behavior, there are two complex frequency dependent interface compliances, ηN and ηT, where the component of the slip normal to the interface is given by ηN times the normal stress and the component tangential to the interface is given by ηT times the shear stress and is in the same direction. Reflection and transmission coefficients for harmonic plane waves incident at arbitrary angles upon a plane linear slip interface are computed in terms of the interface compliances. These coefficients are frequency dependent even when the compliances are real and frequency independent. Examples of the effects of buried slip interfaces on reflection coefficient spectra and on Love‐wave dispersion relations are ...
129440095
Transmission of seismic waves across single natural fractures
Fractures and other nonwelded contacts are important mechanical and hydrological features of rock masses. Their effects on seismic wave propagation can be modeled as a boundary condition in the seismic wave equation. Seismic stress is continuous across such a boundary, but seismic particle displacement and seismic particle velocity are not. The complete solutions for seismic wave reflection, conversion, and transmission across a displacement and velocity discontinuity between two half-spaces with different densities and elastic properties are derived for all angles of the incident wave. The ratio between the seismic stress across this boundary and the seismic particle displacement and velocity are described by a specific stiffness and a specific viscosity, respectively. A displacement discontinuity results in frequency-dependent reflection and transmission coefficients and a frequency-dependent group time delay. The velocity discontinuity results in frequency-independent coefficients and zero delay. Results of laboratory experiments on compressional and shear wave transmission across three different natural fractures in a quartz monzonite are described. Measurements were made at different effective stresses under dry and saturated conditions at room temperature. It is shown that the effect of these fractures on the spectral amplitudes for compressional and shear pulses transmitted across these fractures are described well by a displacement discontinuity for compressional pulses under dry and saturated conditions and by a combined displacement and velocity discontinuity for shear wave pulses under dry and saturated conditions. Values of specific stiffness and specific viscosity vary between fractures and increase with increasing effective stress, as does the static specific stiffness of these fractures. Changes in the spectral amplitudes of transmitted pulses are also analyzed in terms of attenuation using the seismic quality factor Q, which is found to be a function of frequency.
137003416
Study on the transmission and reflection of stress waves across joints
Abstract In order to investigate the transmission and reflection of stress waves across joints, a fractal damage joint model is developed based on fractal damage theory, and the analytical solution for the coefficients of transmission and reflection of stress waves across joints is derived from the fractal damage joint model. The fractal geometrical characteristics of joint surfaces are investigated by using laser profilometer to scan the joint surfaces. The dynamic experiments by using Split Hopkinson Pressure Bar (SHPB) for rock specimens with single joint are conducted to confirm the analytical solution of the coefficients of transmission and reflection. The SHPB experimental results of the coefficients agree well with the analytical solution.
131441751
Analytical Study for Stress Wave Interaction with Rock Joints Having Unequally Close–Open Behavior
Stress wave interaction with rock joints during wave propagation is usually dependent on the dynamic response of the joints. During wave propagation, joints may be closed and open under the effects of the stress wave and the in situ stress. A joint in nature can only resist load during close process. In this paper, the close and open behaviors of rock joints are considered to be different. The joints are assumed to be linearly elastic in close status but turn into free surfaces in open status. Wave propagation equation across joints with unequally close–open behavior is first derived and expressed as a time-differential form based on the displacement discontinuity method. SHPB test recording is then adopted to verify the present approach, which is also compared with the results from existing methods for joints with equally close–open behavior. Next, analysis is conduced for wave propagation across a single joint and a set of parallel joints with unequally close–open behavior, respectively. From the analysis, effects of unequally close–open behavior of a joint on wave propagation and the dynamic response of the joint are studied finally.
127306866
Theoretical study of the dynamic response of a circular lined tunnel with an imperfect interface subjected to incident SV-waves
Abstract Theoretical analysis is used to predict the dynamic response of a circular lined tunnel with an imperfect interface to plane SV-waves. The linear spring model is introduced to depict the imperfect interface between the rock mass and the lining. The expressions for the stress and displacement of the rock mass and the lining are then derived based on the wave function expansion method and the linear spring model. The dynamic stress concentration factor (DSCF) of the rock mass and lining is systematically discussed in two cases by studying its basic parameters, including the dimensionless wavenumber, the interface spring constants, and the lining thickness. The results show that the imperfect interface has a prominent effect on dynamic stress concentration. The maximum values of DSCF generated by SV-waves in the rock mass and the lining are larger than those generated by the P-waves. Additionally, the thick lining structure and tight adhesion of the lining to the rock mass are conducive to tunnel structure stability. It is suggested that the support should be strengthened at the position of the dynamic stress concentration, and the lining and the rock mass should be combined as closely as possible during the construction process.
128826965
UDEC modelling on wave propagation across fractured rock masses
Abstract Wave attenuation across fractured rock masses is a great concern of rock engineers to assess the safety of underground structures in and on rocks under dynamic loads. Due to the discreteness of rock masses, the universal distinct element code (UDEC) has been adopted for the study of rock mass problems. In this paper, the calibration work of UDEC modelling on P-wave propagation across single linearly and nonlinearly deformable fractures is conducted. Subsequently, numerical studies of P-wave propagation across multiple nonlinearly deformable fractures are carried out. The magnitude of transmission coefficient is calculated as a function of nondimensional fracture spacing for different numbers of fractures. The results reveals that under some circumstances, the magnitude of transmission coefficient not only increases with increasing number of fractures, but also is larger than 1.
122324042
Mechanics of Solid Materials
1. Elements of the physical mechanisms of deformation and fracture 2. Elements of continuum mechanics and thermodynamics 3. Identification and theological classification of real solids 4. Linear elasticity, thermoelasticity and viscoelasticity 5. Plasticity 6. Viscoplasticity 7. Damage mechanics 8. Crack mechanics.
129261043
Seismic response of a single and a set of filled joints of viscoelastic deformational behaviour
Rock joints are often filled with weak medium, for example, saturated clay or sand, of viscoelastic nature. Their effects on wave propagation can be modelled as displacement and stress discontinuity conditions. The viscoelastic behaviour of the filled joint can be described by either the Kelvin or the Maxwell models. The analytical solutions for wave propagation across a single joint are derived in this paper by accounting for the incident angle, the non-dimensional joint stiffness, the non-dimensional joint viscosity and the acoustic impedance ratio of the filled joint. It is shown that the viscoelastic behaviour results in dissipation of wave energy and frequency dependence of the reflection and transmission coefficients. Based on curve fitting of the experimental data of P-wave propagation across a single joint filled with saturated sand, both the Kelvin and Maxwell models are found to reproduce the behaviour of the filled joint, in terms of the amplitude and frequency contents. Then, wave transmission across a filled joint set is studied with the virtual wave source method and the scattering matrix method, where multiple wave reflections among joints are taken into account. It is shown that the non-dimensional joint spacing and the number of joints have significant effects on the transmission coefficients.
140579886
Obliquely incident wave propagation across rock joints with virtual wave source method
Due to the presence of joints, waves are greatly attenuated when propagating across rock masses. Zhu et al. (2011) (Normally incident wave propagation across a joint set with virtual wave source method. J. Appl. Geophys.73, 283–288.) studied normally incident wave propagation across a joint set with the virtual wave source method (VWSM). The introduced VWSM has merits in some aspects, especially the capability of separating differently arriving transmitted waves. However, normal wave incidence is only the special case for wave incidence with arbitrary incident angles. Obliquely incident wave propagation across a joint set is more complicated than normally incident wave propagation due to wave transformation at the joints. As a continuation of the previous paper, this work is extended to analytically study obliquely incident wave propagation across joints with VWSM. Complete theoretical reflection and transmission coefficients across single joint described by displacement discontinuity model are derived through plane wave analysis. The superposition of P wave and S wave is for the first time mathematically expressed and studied. The VWSM is verified through comparison with the propagation matrix method. Through extensive parametric studies on wave transmission across single and multiple parallel joints, it is shown that transmitted wave energy is mainly constrained in the transmitted wave of the same type as the incident wave. And with increasing joint stiffness, the transmission coefficients across single joint increases except those whose wave type is different from the incident wave. The amplitude of superposed transmitted wave for P wave incidence increases with incident angle, which is coincident with field observations. Both joint spacing and number of joints have significant effects on transmission coefficients. We find that when joint spacing is sufficiently large, the transmission coefficient is no longer a constant as the normally incident wave propagation case (Zhu et al., 2011). And when joints are very closely spaced, wave attenuation depends little on the number of joints, which is different from the conclusions from equivalent medium method.
134644517
Analytical Time-Domain Solution of Plane Wave Propagation Across a Viscoelastic Rock Joint
The effects of viscoelastic filled rock joints on wave propagation are of great significance in rock engineering. The solutions in time domain for plane longitudinal (P-) and transverse (S-) waves propagation across a viscoelastic rock joint are derived based on Maxwell and Kelvin models which are, respectively, applied to describe the viscoelastic deformational behaviour of the rock joint and incorporated into the displacement discontinuity model (DDM). The proposed solutions are verified by comparing with the previous studies on harmonic waves, which are simulated by sinusoidal incident P- and S-waves. Comparison between the predicted transmitted waves and the experimental data for P-wave propagation across a joint filled with clay is conducted. The Maxwell is found to be more appropriate to describe the filled joint. The parametric studies show that wave propagation is affected by many factors, such as the stiffness and the viscosity of joints, the incident angle and the duration of incident waves. Furthermore, the dependences of the transmission and reflection coefficients on the specific joint stiffness and viscosity are different for the joints with Maxwell and Kelvin behaviours. The alternation of the reflected and transmitted waveforms is discussed, and the application scope of this study is demonstrated by an illustration of the effects of the joint thickness. The solutions are also extended for multiple parallel joints with the virtual wave source method and the time-domain recursive method. For an incident wave with arbitrary waveform, it is convenient to adopt the present approach to directly calculate wave propagation across a viscoelastic rock joint without additional mathematical methods such as the Fourier and inverse Fourier transforms.
128387980
An equivalent viscoelastic model for rock mass with parallel joints
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.