应力波穿越节理时的透反射研究外文文献翻译

英文原文Study on the transmission and reflection of stress waves across jointsYexue Li a,b, Zheming Zhuc,n , Bixiong Lic, Jianhui Denga , Heping Xiec a College of Water Resource & Hydropower, Sichuan University, Chengdu 610065, Chinab Department of Civil Engineering, Xiangfan University, Xiangfan, Hubei 441053, Chinac College of Architecture and Environment, Sichuan University, Chengdu 61005, ChinaAbstract：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.Keywords: Stress wave Transmission and reflection Joint Fractal dimension Damage SHPB1. IntroductionAs is well known, rocks usually contain defects or discontinuities, such as joints, cracks, pores and faults. These discontinuities weaken rock material strength and stability as they are subjected to dynamic loads, such as blasts and earthquakes, therefore, the stability of geotechnical structures, such as tunnels, subways, dams, etc., would be controlled by these discontinuities. The designers of such geotechnical structures are required to have the knowledge of wave propagation across joints, and therefore, the corresponding study on transmission and reflection of waves across joints is significant. Currently, a large number of theoretical, empirical and numerical models concerning the geometrical and mechanical properties of joints have been established to study the joint effects on wave propagation and attenuation 18.However, our understanding of the process of transmission and reflection of stress waves crossing joints is far from complete, as joints are still complex for us.When stress waves encounter joints, they will suffer partial reflection and transmission. The characteristics of reflection and transmission depend on many factors, such as, joint spacing, joint orientation 9, joint width, joint roughness, joint stiffness, filling material inside joints 10,11, liquid saturation, etc. For the issue of wave propagation across joints, Cai and Zhao 4 have investigated the effects of multiple parallel fractures on apparent attenuation of stress wave in rock masses. Fan et al. 6 have presented the study on wave propagation and wave attenuation in jointed rock masses by using discrete element method (DEM). Gulyayev and Ivanchenko 8 have investigated the problem about dynamic interaction of discontinuous waves with interfaces between anisotropic elastic media. Perak-Nolte 12 and Zhao and Cai 13 have investigated stress wave propagation across linear deformation joints, and obtained the coefficients of stress wave transmission and reflection. Wang et al. 14 have derived the coefficients of transmission and reflection of stress wave crossing non-linearly deformational joints by using the rock damage mechanics theory 15 and strain equivalent hypothesis, and the result of the coefficients agreed well with the numerical results presented by Zhao and Cai 13.Kahraman 16 has simulated joint surface roughness and investigated the relation between the roughness of joint surfaces and the particle velocity of stress waves.The surfaces of natural joints developed during a long time of geological digenesis are not smooth, and actually they are very rough. A coefficient of joint roughness JRC (joint roughness coefficient), proposed by Bam ford 17 in 1978, somehow can express the effect of joint roughness on stress wave propagation. However, JRC is an empirical parameter, and it can only express the roughness of one-dimensional curve and cannot be employed to describe the natural joint roughness precisely because the natural joints contain three-dimensional rough surfaces, in which the dimension should be greater than 2 and less than 3. As stress waves cross joints, it is very difficult to determine the incident angle because the joint surfaces usually consist of various sub surfaces with different normal directions. Under this scenario, it is necessary to use another parameter, fractal dimension, to describe the joint roughness. Fractal dimensions can be applied to describing the joint roughness 1820 and it will be employed in this study.In this paper, an attempt is made to obtain the analytical solution for the coefficients of transmission and reflection of stress waves across joints. The fractal theory 2123 combining with damage mechanics theory will be applied to charactering the joint roughness. The fractal geometrical characteristics of joint surfaces are investigated by using laser profilometer. Finally, by using Split Hopkinson Pressure Bar (SHPB) testing system, experimental investigations for the transmission and reflection of stress waves across a single rock joint are conducted.2. Theoretical studyWhen stress waves encounter joints, it is very difficult to determine the incident angle because the joint surfaces usually are very rough, and therefore, the parameters of amplitude and orientation of the transmitted and reflected waves cannot be determined. In the following, we will investigate the factuality of joint surfaces by fractal theory and derive the coefficients of transmission and reflection of stress waves across joints.2.1. Joint stiffnessBased on the fact that the evolution of rock damaging has the characteristic of fractal, Xie and Ju 18 have defined a fractal damage variable(d，) that not only can express the damage intrinsic mechanism quantitatively in two-dimensional Euclidean spaces, but also can be easily adopted in macro-scale damaging analysis by using the theory of damage and fractal d，=1-(1-0)d-de （1）where d is the dimension of the complement of the damage area, is the measurement scale, de is the Euclidean dimension, in 2D case it equals 2, and in 3D case it is 3, and 0 is a nominal damage variable and is defined as 0=SS0=1-SS0 （2）where S0 is an initial area of a zone (the nominal area in Euclidean space), S is the damaged area in the zone, S is the no-damaged area in the zone, and their relation isS0=S +S .Using Esq. (1) and (2), one can obtain the damage variable (d,), and the corresponding theoretical and experimental study has been presented in references 18. However, for the variable 0 expressed in Eq. (2), only the damage occurred in surface planes has been considered, and the damage along depth is ignored. This will somehow cause error as the damage actually varies along depth. Fig. 1 shows a damaged zone with two different damage depths. It can be seen that if one uses Eq. (2) to calculate0, for the two cases, the results will be the same. However, because the damage depths are different, the damage extents should be different too. In order to accurately and effectively describe rock damage, it is necessary to use the ratio of volumes to replace the ratio of areas in Eq. (2), that is =VV0=1-VV0 (3)Where V0 is the initial volume of a zone (the nominal volume in Euclidean space), V is the damaged volume in the zone, V is the no-damaged volume in the zone, and their relation is d 3).Substituting Eq. (3) into Eq. (1), the fractal damage variable can be written as (note: in 3D cases,de=3)(d，)=1-VV0d-3 (4)According to the definition of damage variables, the joint stiffness Kx can be expressed in terms of the damage variable(d，), as Kx=K0(1-(d，) (5)Substituting Eq. (4) into Eq. (5), the joint stiffness of damaged rock joints can be rewritten as Kx=K0VV0d-3 (6)Where K0 is the static normal stiffness of jointsFig. 2 shows two rock specimens; the height for the specimen with joint is H, and the height for the specimen without joint is H-, where is the joint width. Under uniaxial compression, the displacements of both specimens (j for the specimen with joint, and for the specimen without joint) can be easily measured, and the difference, j-, is the joint displacement, thus, the relation of loads P versus joint displacements, j-, can be obtained, and the joint normal stiffness can be calculated.2.2. Coefficients of wave transmission and reflectionAs stress waves encounter joints, they will suffer partial reflection and transmission. In this paper, by using the fractal damage theory and combining with the former analytical solution of wave propagation across straight joints, the coefficients of wave transmission and reflection across joints will be derived. Fig. 3 shows a rough joint and its corresponding equivalent straight joint; the joint stiffness of the straight joint is supposed to be the same as that of the rough joint. The propagation of P waves and S waves are independent, but they are generally mutually relative when either P wave or S wave transmitted and reflected on joints. Except for the case that a wave is normally projected on the interface, that is, normal incidence, two kinds of waves could go through uncoupled solution. For stress waves obliquely incident to joints, in order to meet boundary condition, whether P waves or S waves are sure to simultaneously reflect P waves and S waves, and transmit P waves and S waves.Considering the case that only a plane P wave incident on the straight joint interface, which will produces simultaneously a reflected P wave and S wave, and a transmitted P wave and S wave as shown in Fig. 3. For each of these five waves, there is a corresponding wave propagation equation, thus, there are totally five equations which can be expressed as follows 24:u0=A0expiKx0x+ky0y-tu1=A1expiKx1x+ky1y-tu2=A2expiKx2x+ky2y-tu3=A3expiKx3x+ky3y-tu4=A4expiKx4x+ky4y-t (7)where u0，u1，u2，u3，u4 are the displacements induced by the incident P wave, reflected P wave, reflected S wave, transmitted P wave and transmitted S wave, respectively, A0, A1, A2, A3, A4 are the corresponding wave amplitudes, respectively, and and k are the angular frequency and wave vector, respectively. The displacement components in zone I and zone II as shown in Fig. 3 can be expressed asux1=u0cos1-u1cos1+u2sin1uy1=u0sin1+u1sin1+u2cos1 (8)ux2=u3cos2-u4sin2uy2=u3sin2+u1cos2 (9)where 1 ，2， 1 and 2 are the angles shown in Fig. 3, ux1，uy1，ux2，uy2are the displacement components along x and y axial direction, respectively, the subscripts 1 and 2 in Esq. (8) and (9) represent zone I and zone II, respectively. Supposing the deformations induced by the stress waves are in the elastic range, then, the relation between displacement and stress can be written asxx=(+2)uxx+uyy xy=(uyx+uxy) (10)Where is shear modulus According to the displacement discontinuity model proposed by Perak-Nolte 12, the stresses on the common boundary of zone I and zone II satisfy the following Relationship:xx1=xx2 (11) xy1=xy2 (12)ux1-ux2=xxKx (13)uy1-uy2=xyKy (14)Where Kx and Ky are the joint normal stiffness and shear stiffness, respectively. The coefficients of transmission and reflection are defined asFR1=A1A0， FR2=A2A0 (15)FT1=A3A0， FT2=A4A0 (16)Where FR1 and FR2 are the reflection coefficients of the reflected P wave and S wave, respectively, and FT1 and FT2 denote the transmission coefficients of the transmitted P wave and S wave, respectively. As stress waves propagate in an elastic body, the relation of the parameters can be expressed as+2=P2 (17)=S2 (18)=P2-2S2 (19)Where=E1+1-2, is the Poissons ratio, Cp is the P wave speed, Cs is the S wave speed, is the shear modulus and is the density. From Esp. (7) to (19), we obtain the following equations:s12p1sin21-FR1s12p1sin21-FR2s1cos21-21FT1s22p2sin22+FT2s2cos22=0 (20) p1-sin212s12p11+FR1-FR2s1sin21-21FT1p2-sin222s22p2-FT2s2sin22=0 (21)cos1-FR1cos1+FR2sin1-FT1cos2-FT2sin2=p1-sin212s12p11+FR2-FR2s1sin21Kxi1 (22)sin1+FR1sin1+FR2cos1-FT1sin2+FT2cos2=s12p1sin21-FR1s12p1sin21-FR2s2cos22Kyi1 (23)Where are wave angular frequency, and the subscripts 1 and 2 in the above equations represent zone I and zone II, respectively. For normal incidence, i.e. the incident P wave is perpendicular to the joint surfaces, one can have1=2=1=2=0. If the rocks properties at both sides of the joint are the same, then the density of rock mass and the velocity of stress wave at both sides are identical, thus, from Esq. (20) to (23), we obtain the coefficients of the reflected P wave and the transmitted P wave asFR1=11+4KxZ2， FT1=2Kx/Z1+4KxZ2 (24)Where :Z is the wave impedance. Substituting Eq. (6) into Eq. (24), we obtain the analytical solution of the coefficients of transmission and reflection as stress waves cross jointsFR1=11+4K0VV0d-32/Z2FR1=2K0ZVV0d-31+4K0VV0d-32/Z2 (25)3. Experiment studyIn order to validate the theoretical solution, i.e. Eq. (25), dynamic experimental study for rock specimens with a single joint by using Split Hopkins Pressure Bar (SHPB) is conducted, and by using a laser profilometer, the joint surfaces have been scanned, and the joint surface fractal dimensions have been measured, and the results have been employed to calculate the coefficients of transmission and reflection of waves across joints.3.1. SHPB testing systemSplit Hopkins Pressure Bar is widely used for characterizing dynamic response of engineering materials. The SHPB system used in this study is shown in Fig. 4.The lengths of the incident bar and transmission bar of the SHPB are the same, 2.0 m, and their diameters are 32 mm. To minimize the unexpected effects of inertia, the transversal dispersion and the friction between the ends of the specimens and the input and output bars, we applied a single jointed specimen with 12 mm in length and 30 mm in diameter. The incident impact waves were generated and propagated along the axial directions of the specimen cylinders, and perpendicularly projected to the joint surfaces, which is located in the middle of the 12 mm long specimen. The objective of this experimental study is to investigate the effect of joint surface configuration on wave reflection and transmission, so as to validate the theoretical results of the coefficients of wave reflection and transmission. In order to minimize the side effect of large plastic deformation and additional cracking, the impact striker speed should be controlled within a certain range, such that the plasticity and cracking in the specimen induced by the impact were negligible, thus, no irreversible energy will dissipate except for the displacement of joint surface during the wave propagation. According to the preliminary SHPB tests using intact rocks with different impact speed, the impact speed of 6.8 m/s was adopted finally in the SHPB tests.The incident wave, reflected wave and transmitted wave are collected by super dynamic strain apparatus. Stress gauges were employed in recording the stress waves, and were situated in the middle of the incident and transmission bars, respectively (see Fig. 4). In order to record the stress waves continuously and accurately and to avoid the inuence of pulses reflected from the free and the contacted ends, the distance between the strain gauges and the ends of the bars is larger than the length of the striker bar which is 200 mm in length.3.2. SHPB specimen preparation and testing resultsRock cores with a diameter of 30 mm were first drilled from marble blocks. To avoid the effect of natural micro-joint on experiment result, the parts of the rock cores with compact structure were selected to make rock specimens. By three-point bending method, the selected rock cores were fractured into two parts,and these two parts were adhered together and by cutting the two ends of the core, a 12 mm long specimen with a joint in themiddle, shown in Fig. 5, was made. The ends of the specimens have been ground very carefully such that the frictionless condition can be satisfied before we installed the specimen between the steel bars. Because the specimen length 12 mm is very short as compared to the incident bar, the strains on the two ends are approximately the same in a very short time interval, which can meet the hypothesis of strain uniformity. Because the length of the specimens satisfies the requirement of h=3r, where r is the radius of the specimen cylinders and n is the dynamic Poisso