Three-dimensional finite element simulation of fracture propagation in rock specimens with pre-existing fissure(s) under compression and their strength analysis

A FORTRAN program, consistent with the commercially available finite element (FE) code ABAQUS, is developed based on a three-dimensional (3D) linear elastic brittle damage constitutive model with two damage criteria. To consider the heterogeneity of rock, the developed FORTRAN program is used to set the stiffness and strength properties of each element of the FE model following a Weibull distribution function. The reliability of the program is assessed against available experimental results for granite cylindrical specimens with a throughgoing, flat and inclined fissure. The calibration procedure of the material parameters is explained in detail, and it is shown that the compressive to tensile strength ratio can have a substantial influence on the failure response of the specimens. Numerical simulations are conducted for models with different levels of heterogeneity. The results show a smaller load bearing capacity for models with less homogeneity, representing gradual coalescence of fully damaged elements forming throughout the models during loading. The maximum load bearing capacity is studied for various combinations of inclination angles of two centrally aligned, throughgoing and flat fissures of equal length embedded in cylindrical models under uniaxial and multiaxial loading conditions. The key role of the compressive to tensile strength ratio is highlighted by repeating certain simulations with a lower compressive to tensile strength ratio. It is proven that the peak loads of the rock models with sufficiently small compressive to tensile strength ratios containing two throughgoing fissures of equal length are similar, provided that the minimum inclination angles of the models are the same. The results are presented and discussed with respect to the existing experimental findings in the literature, suggesting that the numerical model applied in this study can provide useful insight into the failure behaviour of rock-like materials.     

Implementation of a coupled plastic damage distinct lattice spring model for dynamic crack propagation in geomaterials

This paper studies dynamic crack propagation by employing the distinct lattice spring model (DLSM) and 3‐dimensional (3D) printing technique. A damage‐plasticity model was developed and implemented in a 2D DLSM. Applicability of the damage‐plasticity DLSM was verified against analytical elastic solutions and experimental results for crack propagation. As a physical analogy, dynamic fracturing tests were conducted on 3D printed specimens using the split Hopkinson pressure bar. The dynamic stress intensity factors were recorded, and crack paths were captured by a high‐speed camera. A parametric study was conducted to find the influences of the parameters on cracking behaviors, including initial and peak fracture toughness, crack speed, and crack patterns. Finally, selection of parameters for the damage‐plasticity model was determined through the comparison of numerical predictions and the experimentally observed cracking features.     

Analytical study of ground motion caused by seismic wave propagation across faulted rock masses

Studying seismic wave propagation across rock masses and the induced ground motion is an important topic, which receives considerable attention in design and construction of underground cavern/tunnel constructions and mining activities. The current study investigates wave propagation across a rock mass with one fault and the induced ground motion using a recursive approach. The rocks beside the fault are assumed as viscoelastic media with seismic quality factors, Q_p and Q_s. Two kinds of interactions between stress waves and a discontinuity and between stress waves and a free surface are analyzed, respectively. As the result of the wave superposition, the mathematical expressions for induced ground vibration are deduced. The proposed approach is then compared with the existing analysis for special cases. Finally, parametric studies are carried out, which includes the influences of fault stiffness, incident angle, and frequency of incident waves on the peak particle velocities of the ground motions.     

P‐wave transmission across fractures with nonlinear deformational behaviour

Stress wave attenuation across fractured rock masses is a great concern of underground structure safety. When the wave amplitude is large, fractures experience nonlinear deformation during the wave propagation. This paper presents a study on normal transmission of P‐wave across parallel fractures with nonlinear deformational behaviour (static Barton–Bandis model). The results show that the magnitude of transmission coefficient is a function of incident wave amplitude, nondimensional fracture spacing and number of fractures. Two important indices of nondimensional fracture spacing are identified, and they divide the area of nondimensional fracture spacing into three parts (individual fracture area, transition area and small spacing area). In the different areas, the magnitude of transmission coefficient has different trends with nondimensional fracture spacing and number of fractures. In addition, the study reveals that under some circumstances, the magnitude of transmission coefficient increases with increasing number of fractures, and is larger than 1. Copyright © 2006 John Wiley & Sons, Ltd.     

Plane of elastic non-resisting tension material under foundation structures

In the frame of 2D-static problems one approaches the problem of elastic-NRT (not-resisting tension) semi-plane loaded on its limit line. This problem is intended to model the stress situation induced in the soil by a foundation structure. The solution, in terms of activated stress field, is searched for in the class of stress fields satisfying equilibrium and admissibility conditions, by applying an energy approach. Copyright © 2004 John Wiley & Sons, Ltd.     

The response of an elastic half‐space under a momentary shear line impulse

The response of an ideal elastic half‐space to a line‐concentrated impulsive vector shear force applied momentarily is obtained by an analytical–numerical computational method based on the theory of characteristics in conjunction with kinematical relations derived across surfaces of strong discontinuities. The shear force is concentrated along an infinite line, drawn on the surface of the half‐space, while being normal to that line as well as to the axis of symmetry of the half‐space. An exact loading model is introduced and built into the computational method for this shear force. With this model, a compatibility exists among the prescribed applied force, the geometric decay of the shear stress component at the precursor shear wave, and the boundary conditions of the half‐space; in this sense, the source configuration is exact. For the transient boundary‐value problem described above, a wave characteristics formulation is presented, where its differential equations are extended to allow for strong discontinuities which occur in the material motion of the half‐space. A numerical integration of these extended differential equations is then carried out in a three‐dimensional spatiotemporal wavegrid formed by the Cartesian bicharacteristic curves of the wave characteristics formulation. This work is devoted to the construction of the computational method and to the concepts involved therein, whereas the interpretation of the resultant transient deformation of the half‐space is presented in a subsequent paper. Copyright © 2003 John Wiley & Sons, Ltd.     

Seismic rays in an ellipsoidal earth model: theoretical analysis and estimation of deviations due to the ellipticity

Summary. A first-order form of the Euler's equations for rays in an ellipsoidal model of the Earth is obtained. The conditions affecting the velocity law for a monotonic increase, with respect to the arc length, in the angular distance to the epicentre, and in the angle of incidence, are the same in the ellipsoidal and spherical models. It is therefore possible to trace rays and to compute travel times directly in an ellipsoidal earth as in the spherical model. Thus comparison with the rays of the same coordinates in a spherical earth provides an estimate of the various deviations of these rays due to the Earth's flattening, and the corresponding travel-time differences, for mantle P -waves and for shallow earthquakes. All these deviations are functions both of the latitude and of the epicentral distance. The difference in the distance to the Earth's centre at points with the same geocentric latitude on rays in the ellipsoidal and in the spherical model may reach several kilometres. Directly related to the deformation of the isovelocity surfaces, this difference is the only cause of significant perturbation in travel times. Other differences, such as that corresponding to the ray torsion, are of the first order in ellipticity, and may exceed 1 km. They induce only small differences in travel time, less than 0.01s. Thus, we show that the ellipticity correction obtained by Jeffreys (1935) and Bullen (1937) by a perturbational method can be recovered by a direct evaluation of the travel times in an ellipsoidal model of the Earth. Moreover, as stated by Dziewonski & Gilbert (1976), we verify the non-dependence of this correction on the choice of the velocity law.     

Observations of unusual whistlers during daytime at Jammu

In this paper, we report observations of unusual whistlers recorded at Jammu (geomag. lat. = 22°26′N; L = 1.17), India on March 8, 1999 during the daytime. They are interpreted as one-hop ducted whistlers having propagated along higher L-values in closely spaced narrow ducts from the opposite hemispheres. After leakage from the duct, the waves might have propagated in the earth-ionosphere waveguide towards the equator in surface mode. Tentative explanation of the dynamic spectra of these events is briefly presented.     

水库库岸滑坡涌浪的传播与爬高研究

基于流体力学明渠非恒定流的连续性方程、运动方程和沿程水头损失理论,把滑坡涌浪衰减过程分为急剧衰减和缓慢衰减两个阶段来考虑,并认为急剧衰减阶段的涌浪的衰减符合指数衰减规律,缓慢衰减阶段符合明槽水流的沿程水头损失规律,结合初始涌浪高度对涌浪沿岸的传播高度及爬坡高度进行了计算。以新滩滑坡为例对涌浪的传播及爬坡高度进行了计算,得出了滑坡涌浪衰减先快后慢以及传播3 km时的涌浪高度只有初始涌浪高度的30 %以及传播10 km时涌浪高度只有初始涌浪高度的13 %的规律。