Discontinuous Deformation Analysis Method Research Articles

Implementation of a J-integral based Maximum Circumferential Tensile Stress theory in DDA for simulating crack propagation

Crack propagation is a very important research subject within the field of rock mechanics. In this study, a J-integral based Maximum Circumferential Tensile Stress theory was applied to the conventional Discontinuous Deformation Analysis (DDA) Method for simulating crack propagation. First, according to the relevant theories of fracture mechanics, a crack propagation theory suitable for DDA was derived by calculating J-integral in DDA simulation. Then, the crack propagation algorithm was embedded in the DDA algorithm with an accurate judgement of the propagation direction. After that, the numerical calculation results were compared with the experimental results to verify the correctness of the crack propagation algorithm. The validated method was finally applied to simulate a landslide case with a preexisting crack. The results showed that the proposed method was applicable to simulating crack propagations in practical cases.

Modeling Wave Propagation in Rock Masses Using the Contact Potential-Based Three-Dimensional Discontinuous Deformation Analysis Method

The most recently proposed three-dimensional (3D) contact potential-based discontinuous deformation analysis (3D-CPDDA) method is further applied for wave propagation problems in rock masses. A viscous non-reflecting boundary is incorporated into the 3D-CPDDA method to minimize the wave reflections. Furthermore, a force input method is incorporated to eliminate the contamination of scattered waves in the numerical solution and to accurately input the incident wave. Several benchmark problems about P-wave/S-wave propagation in homogeneous rock masses and jointed rock masses are solved to validate the modified 3D-CPDDA method. The numerical results assessed by the modified 3D-CPDDA method agree well with those obtained from analytical methods, which means that the modified 3D-CPDDA method can reliably and correctly simulate wave propagation in rock masses. The modified 3D-CPDDA method warrants further investigation.

Investigations into the Rock Dynamic Response under Blasting Load by an Improved DDA Approach

Evaluation of blasting-induced rock damage and fragmentation is very important for safety control of construction in the jointed rock mass. The discontinuous numerical models are commonly applied due to the advantages in modeling fragmentation and treating discontinuities. In this paper, the rock fracturing algorithm and rate dependent strength law are incorporated into the discontinuous deformation analysis (DDA) to study the wave propagation and rock fragmentation phenomena in dynamic problems. By reproducing Hopkinson pressure bar tests under different loading rates, the improved method is validated to be capable of solving dynamic failure problem. Finally, taking the Xiucun tunnel as an example, its failure process under the action of the explosive wave is simulated, and some failure features are captured. In addition, the explosion wave propagation and its induced particle vibration in surrounding rock are studied. The reasonable simulation results indicate that the modified DDA method can effectively model the stress wave propagation and joint growth process in the jointed rock under blasting load.

Numerical modeling of complex hydraulic fracture networks based on the discontinuous deformation analysis (DDA) method

Hydraulic fracturing is one of the most important technologies for shale gas production. Complex hydraulic fracture networks can be stimulated in shale reservoirs due to the existence of numerous natural fractures. The prediction of the complex fracture network remains a difficult and challenging problem. This paper presents a fully coupled hydromechanical model for complex hydraulic fracture network propagation based on the discontinuous deformation analysis (DDA) method. In the proposed model, the fracture propagation and rock mass deformation are simulated under the framework of DDA, and the fluid flow within fractures is simulated using lubrication theory. In particular, the natural fracture network is considered by using the discrete fracture network (DFN) model. The proposed model is widely verified against several analytical and experimental results. All the numerical results show good agreement. Then, this model is applied to field-scale modeling of hydraulic fracturing in naturally fractured shale reservoirs. The simulation results show that the proposed model can capture the evolution process of complex hydraulic fracture networks. This work offers a feasible numerical tool for investigating hydraulic fracturing processes, which may be useful for optimizing the fracturing design of shale gas reservoirs.

Verification of a DDA-based hydro-mechanical model and its application to dam foundation stability analysis

There are many studies concerning hydro-mechanical coupling model based on discontinuous deformation analysis (DDA). However, verification of the coupling model and its application to practical engineering are rarely found. In this paper, we first introduced the distributed load sub-matrix of water pressure and described the calculation procedures of hydro-mechanical coupling analysis in the framework of DDA. Then, the correctness of the coupling model was verified respectively through analyses of buoyancy, deformation of a block under uniform water pressure and hydraulic fracturing. Finally, the model was applied to evaluate the sliding stability of a gravity dam foundation in China. The failure process and stability factor were calculated by two different stability evaluation methods (the strength reduction method and the overloading method), and the obtained stability factors were compared with those solved by the commonly used limit equilibrium analysis methods. The strength reduction method revealed that the rock strata of dam foundation slide with different displacements along several weak interlayers of different depths, whereas the overloading method revealed that the rock strata slide mainly along a special interlayer beneath the anti-sliding concrete stud, which was designed to be near the dam heel. The DDA method clearly shows the process of rock block failure and profoundly reveals the difference in sliding mode caused by using different stability evaluation methods.

Improvement of DDA with a New Unified Tensile Fracture Model for Rock Fragmentation and its Application on Dynamic Seismic Landslides

Discontinuous deformation analysis (DDA) method is a discrete element method, presenting a great advantage in modelling deformation and rigid body movements, and it is also an alternative approach for problems involving the fracturing process from continuity to discontinuity if the failure mechanism in DDA is well constituted. This paper presents a new united tensile fracture model (UTFM) for the two-dimensional DDA method to simulate the fracture behaviors of various brittle materials (e.g., rock, soil, and concrete). The new fracture model unifies four classical failure modes, including the maximum normal stress criterion, Tresca criterion, Mohr–Coulomb criterion, and the von Mises criterion, for tensile fracture. By incorporating UTFM into the original DDA frame, the improved DDA (I-DDA) can predict the crack initiation and propagation paths in Brazil disc and simulate rock fracture of various brittle materials. Numerical examples of the direct tensile test and the Brazil disc split tests are investigated to verify the accuracy and validity of the I-DDA method. The simulated results agree well with those obtained from physical tests and other numerical analyses, suggesting that the I-DDA has obvious advantage in simulating the fracture behaviors of the Brazil disc split test. Further, the I-DDA is applied to analyze the failure process of a practical earthquake-induced landslide with consideration of the tensile strength of the rock mass. The results indicate that the I-DDA is more feasible to analyze the slope failure, which can consider both the tensile and shear characteristics simultaneously compared with the original DDA.

A possible mechanism of earthquake-induced landslides focusing on pulse-like ground motions

This paper proposes a mechanism of earthquake-induced landslides with pulse-like ground motion (PLGM) based on the discontinuous deformation analysis (DDA) method. In recent near-fault earthquake-induced landslides, the two phenomena confusing landslide researchers are observed. One is that large-scale landslides occurred in the area with small PGA but no landslide in the area with large PGA, for example, the landslides occurred during the 2016 Kumamoto earthquake. The other is that one side slopes collapsed while the opposite side slopes back to back remain stable. For example, most west-faced slopes collapsed while the east-faced slopes back to back remain stable in the 2018 Hokkaido earthquake. To explain the two phenomena in near-fault earthquake-induced landslides with PLGM, a symmetrical slope model subjected to different ground motions during these two earthquakes are analyzed and discussed by using the DDA method. The analysis results show that PLGM may be a major triggering factor for co-seismic landslides in the near-fault region, while PGA may not be the right parameter for the initiation of the earthquake-induced landslides. Meanwhile, the aspect of collapsed slope is related to both the velocity pulse of PLGM and strength parameters of the slope. The proposed mechanism for the earthquake-induced landslides in the near-fault region with PLGM considers both tensile and shear strengths of the slope, which can well explain the landslide initiation phenomena in the 2016 Kumamoto earthquake and the 2018 Hokkaido earthquake. The relationship between PLGM and forward directivity effect is also studied in combination with two earthquake cases to more accurately clarify the mechanism of the initiation of co-seismic landslides in the near-fault region.

Motion characteristics of rockfall by combining field experiments and 3D discontinuous deformation analysis

Rockfall is a common geological disaster in rock slope projects. An understanding of the motion processes and characteristics of the rock blocks originating from a rockfall is of profound importance for comprehending the mechanism(s) of rockfall disasters. In this study, a field experimental system comprising binocular stereovision was developed for examining rockfalls, and a series of experiments on rockfall motion characteristics were performed on slopes near an abandoned section of the G318 national highway in Tibet, China. The indices of motion characteristics such as the lateral displacement, staying position, jumping height, and kinetic energy evolution of the blocks under different work conditions (e.g. different masses, shapes, falling heights, falling angles, and different geometrical characteristics in the slopes) were investigated by combining experiments with a three-dimensional discontinuous deformation analysis (3D DDA) method. The results show that the field experimental system has high accuracy in monitoring the block movement processes. The 3D DDA results agree well with those from the experiments, and the evolutions of the motion characteristics of the rockfall under different conditions are quantitatively obtained. Furthermore, the results provide insight into the laws of rockfall movements and dynamic impact processes, and lay a foundation for the design of rockfall protection measures and engineering approaches for disaster prevention and mitigation.

Study on the dynamic response of dip bedded rock slope using discontinuous deformation analysis (DDA) and shaking table tests

AbstractStability prediction for bedded rock slopes under seismic loading is very important for landslide hazard assessment. To clarify the dynamic behaviors and damage mechanism of dip bedded slope, shaking table tests and its prototype simulation analyses were performed. Shaking table tests were conducted to analysis the acceleration dynamic response firstly and then the discontinuous deformation analysis (DDA) method was used to reproduce the test results, and study the influence of vertical earthquake on the dynamic response of dip bedded rock slope. Both the two methods results reveal that the amplification coefficients increase with increasing acceleration amplitude under sinusoidal waves shaking condition, and show an increased trend with the relation height. With Wenchaun waves loading, the discontinuous characteristics between the test model and numerical model are consistent, the dynamic responses of the slope tops are strong, and it is easy to be destroyed. Compare the failure mechanism of the slope between shaking table test and numerical method, the damage areas both the numerical model and test model are the nearly identical, and the failure modes of two models are mainly the cracking sliding failure. Moreover, with horizontal and vertical earthquake loading, the acceleration response on the numerical slope top is strong, and the maximum peak acceleration amplification factors are at the leading and rear edges of the slope top. In overall, the dynamic response of the slope surface is more significant than that inside the slope. However, with an increase of amplitude, acceleration responses are different with different directional loading, the acceleration response with horizontal loading is stronger than with vertical loading.

Numerical investigation on the hydraulic stimulation of naturally fractured Longmaxi shale reservoirs using an extended discontinuous deformation analysis (DDA) method

The Silurian Longmaxi formation in the Sichuan Basin, which is the most important gas-producing shale reservoir in China, is characterized by relatively small effective thickness, large in-situ stress difference and highly-developed natural fracture (NF) system. Microseismic monitoring shows that complex hydraulic fracture (HF) networks with relatively small height could be stimulated in the Longmaxi shale reservoirs. Traditional two-dimensional (2D) numerical models cannot capture the three-dimensional (3D) nature of the limited-height HFs, which is not suitable for investigating the fracturing of Longmaxi shale reservoirs. This work develops a fully coupled hydro-mechanical model for HF network propagation based on the discontinuous deformation analysis (DDA) method. A new efficient method is proposed to consider the 3D effect in 2D DDA-based fracturing simulations. Hence an approximate 3D solution of fracture-induced stress and deformation could be obtained. This method is verified against previous works. Then, a series of simulations is performed to investigate the influence of reservoir thickness, in-situ stress difference and NF pattern on the hydraulic stimulation of Longmaxi shale reservoirs. Modeling results show that limited reservoir thickness is favorable to the formation of HF network. Complex and dense fracture network is easily to be formed in thin reservoirs. Large in-situ stress difference will impede the hydraulic stimulation. However, when the reservoir thickness is small, dense fracture network is still likely to be formed under large stress differences. Besides, the NF system is essential for the formation of HF network. The modeling results indicate that as a result of the limited thickness and highly-developed NF system, complex and dense fracture networks could be stimulated in the Longmaxi shale reservoirs. This work partly accounts for the initial success of gas recovery in the Longmaxi shale reservoirs. It also provides a feasible numerical method and theoretical supports for further optimization of shale gas exploitation in China.

Numerical Analyses on the Rockbolt Behaviour in Rock Wedges Reinforcement Using 2D-DDA

One major concern in the design and construction of caverns in jointed rock mass is the potential failure of rock wedges falling from the roof or sliding out from the sidewalls. The rockbolts are commonly used to improve the stability and to prevent failure in the caverns. In this paper, the two-dimensional discontinuous deformation analysis (2D-DDA) method was used to analyse the stability of the rock wedge supported by rockbolts on the cavern roof. The joint relaxation method was adopted to represent a deformable rock joint under stresses. Parametric studies were conducted to investigate the effects of the confining pressure, the spacing and the angle between rockbolt and direction of wedge falling on the stability of the rock wedge. The numerical simulation results indicate that the wedge is self-supporting after stress redistribution if the confining pressure is larger than the critical confining pressure and the semi-apical angle of the wedge is less than the friction angle of rock joint. The force in rockbolt slightly decreases with reduced rockbolts spacing but increase when the rockbolt is not along the falling direction of rock wedge.

Angle-Based Contact Detection in Discontinuous Deformation Analysis

This study presents a novel angle-based contact detection method to address the concave contact problem of arbitrary polyhedra, including convex and concave polyhedra, those with cavities and/or holes, and likely their unions. First, the most important mathematical concept in this study, the angle, is introduced to represent a general polyhedron. Using angles, the topologies of the polyhedra can be far more complex and universal than those previously studied, allowing for the coplanarity of faces and collinearity of edges, and not limiting the polyhedra to simple homeomorphic to closed three-dimensional (3-D) spheres. Second, all the local entrances of two general angles, which are either vertex angle to half-space or crossing edge angle to edge angle entrances, are identified using the entrance formulas. Third, local convex decomposition (LCD) is proposed to decompose any arbitrarily concave angle into a set of convex subangles, making the method easy to implement and naturally compatible with detection of entrance angles. Fourth, the proposed entrance angle method (EAM) is implemented in 3-D discontinuous deformation analysis (DDA) to compute arbitrarily convex/concave contacts. Finally, the EAM-based 3-D DDA method is validated and then employed to investigate the discontinuous mechanical behaviors of complex polyhedral block systems. Overall, the extended 3-D DDA program is sufficiently developed to meet the analysis requirements of complex rock mass projects.

Analysis of landslide stability under seismic action and subsequent rainfall: a case study on the Ganjiazhai giant landslide along the Zhaotong-Qiaojia road during the 2014 Ludian earthquake, Yunnan, China

In a strong earthquake, not only a large number of coseismic landslides are triggered, but relaxation and cracks in the rocks and soils are induced, which make these rocks and soils vulnerable to instability during subsequent rainfall; thus, strong earthquakes always have long-term effects on landslides. The geo-hazards along the Zhaotong-Qiaojia road in the 2014 Ludian earthquake (Ms 6.5) of Yunnan Province, China, were investigated, and the Ganjiazhai giant landslide was chosen as a case study. First, using the limit equilibrium analysis and Newmark method, the critical seismic intensity of the landslide before the earthquake was evaluated. Secondly, the dynamic failure process of the landslide under the measured ground motion was simulated with the discontinuous deformation analysis method. Lastly, based on stress–seepage coupling analysis and precipitation data from Ludian meteorological station, the stability of the landslide during subsequent rainfall after the earthquake was predicted. The results show that the critical seismic intensity was within degrees VIII–IX, which is consistent with the results of the earthquake damage investigation. The dynamic failure process can be divided into four stages, and four scarps formed; the potential sliding zones during the subsequent rainfall were at the first scarp and the fourth scarp, and their critical rainfall amounts were 35–40 mm and 55–60 mm, respectively. In this paper, failure process simulation and stability prediction of the landslide before, during, and after the strong earthquake are presented, which provide analysis methods for the dynamic stability of landslides in the meizoseismal area.

Verifying discontinuous deformation analysis simulations of the jointed rock mass behavior of shallow twin mountain tunnels

This study extends discontinuous deformation analysis (DDA) to simulate the interactions of twin tunnels in a jointed rock mass. The ground subsidence and the rock stress distribution around the tunnels from the DDA simulation are compared to those from the trap-door model. Both the experimental and computational results show that the changing tunnel interactions in the jointed rock mass between the two tunnels are coincidently associated with the tunnel distance. In addition, the dip angle of the rock strata significantly affects the surface subsidence patterns and the stress distribution around the excavation zone. Therefore, the DDA method is a useful tool for describing the mechanical behavior of a jointed rock mass when a twin-tunnel is constructed in mountain terrain.

A study on the false volume expansion problem under rigid large rotation based on the modified symmetric and anti-symmetric decomposition method

During the simulation of large rotation deformation by the discontinuous deformation analysis method or numerical manifold method, a false volume expansion problem appears. A theoretical and experimental study on the false volume expansion problem based on the modified symmetric and anti-symmetric decomposition (MSAD) is given, reveals the intrinsic cause, and gives an insight into the false volume expansion problem. It is theoretically proven that MSAD will never lead to the false volume expansion problem, and it is also verified by two numerical experiments of large rotation deformation in both static and dynamic situations. The MSAD method can accurately distinguish rotation and strain from the incremental deformation gradient tensor and realize the objective update of rotational stress. Using the MSAD method, larger rotational deformation is allowed for each incremental step; thus, fewer calculating steps and higher computational precision can be achieved, indicating that large rotational deformation problems can be solved in a more accurate and efficient manner. The combination of the MSAD method and the Bathe implicit time integration scheme provides good numerical stability and high computational precision and satisfies the conservation of kinetic energy and momentum for conservative systems.

A new contact potential based three-dimensional discontinuous deformation analysis method

The three-dimensional discontinuous deformation analysis (3D-DDA) method was developed for the deformation simulation of rock block system cut by the natural discontinuities in rock mass engineering. In the conventional DDA, open-close iteration is used to deal with the contact constraints, which needs to apply or remove the normal or tangential springs repeatedly to meet the equilibrium equations at each time step. DDA provides a time step adjustment strategy to meet the fast convergence of open-close iterations, but when solving large-scale problems, the adjusted time step often reaches a very small order of magnitude, which makes the calculation time-consuming increase sharply. In the framework of the original 3D-DDA, a new contact potential based three-dimensional discontinuous deformation analysis method (3D-CPDDA) is developed. The proposed method not only retains the advantage of the original DDA method in defining local displacement functions on a single patch, but also integrates the simplicity and rapidity of potential based contact processing. The improved method is easier to be implemented in the parallel way, which can further improve the computational efficiency. Numerical examples have confirmed the correctness and feasibility of the proposed procedure.

Numerical study of ballast-flight caused by dropping snow/ice blocks in high-speed railways using Discontinuous Deformation Analysis (DDA)

Abstract Frozen snow/ice blocks drop at high speeds from train causing the ballast to fly up and damage the car body. Thus, in this paper, we propose a numerical model based on the discontinuous deformation analysis (DDA) method to study the ballast flight caused by dropping snow/ice blocks in high-speed railways and to analyze the dynamic behavior of ballast particles during their collision with a snow/ice block. The validation of the proposed model is done by comparing the numerical results with the theoretical and the experimental ones. The numerical results show that the velocity, shape and incident angle of snow/ice block play an important role in the ballast flight. Specifically, the number and the maximum displacement of ballast particles increase as the train speed increases and the incident angle greatly affects the movement direction of ballast particles. The shape of the ice block affects the amount and extent of ballast flight.

Influence of “X” Fracture, Confining Pressure, and Temperature on Rock Masses’ Failure Process

Based on the similarity theory, sandstone was taken as the prototype, and rock‐like specimens were made with the strength ratio of 1 : 1. Single “X” fracture and double “X” fractures were prefabricated in rock‐like specimen, and crack propagation was studied through the compressive test. An improved discontinuous deformation analysis method (DDARF) was adopted to simulate on the cracking process. Further, other factors should not be ignored such as confining pressure and temperature, which were considered: rock’s crack propagations under loading and unloading with different confining pressures were studied; influences of temperature from 20°C to 300°C on crack propagation were analyzed.

A modified sub-block DDA fracturing modelling method for rock

Discontinuous deformation analysis (DDA) computes the mechanical behaviours of discrete deformable-block systems. It also has been used to simulate continuous rock fracturing, e.g., through the sub-block approach. In the present study, the sub-block DDA method is first verified in simulations of continua. The influences of the contact penalty spring stiffness and mesh size on the stress and strain calculations are investigated. Thereafter, the fracturing modelling algorithm of the sub-block DDA is improved, which determines the tensile or shear fracturing failure along artificial joints based on the stress state of adjacent sub-blocks according to the maximum tensile stress criterion and the Mohr–Coulomb criterion, respectively. The tensile, shear, mixed-mode, as well as dynamic fracturing failures of rock samples are simulated, and the results are calibrated theoretically or experimentally. Compared with former sub-block DDA fracturing modelling algorithms which determine the fracturing failure along artificial joints based on the contact stresses between sub-blocks, the improved algorithm has better accuracy in terms of the failure strength, and it significantly reduces the influence of the distribution of the pre-set artificial joints on the failure strength and fracturing route simulation results. This work makes DDA a better candidate for use in rock fracturing problem simulations.

Runout simulation of seismic landslides using discontinuous deformation analysis (DDA) with state-dependent shear strength model

The reliable numerical simulation of the landslide process contributes to the establishment of evidence-based disaster mitigation measures in seismically active zones. To achieve this goal, a simple and unified state-dependent shear strength model of discontinuities is presented to describe the shear strength degradation in a seismic landslide process. The proposed model establishes a relationship between the shear strength parameters and the global safety factor of the slope by assuming that the slope instability (or landslide initiation) is accompanied by an instantaneous shear strength degradation of discontinuities. To realize the model numerically, the algorithms for the computation of global safety factor and the modification of shear strength parameters were incorporated into the discontinuous deformation analysis (DDA). Subsequently, the kinematic accuracy of the improved DDA method was validated by comparisons with theoretical solutions for the dynamic sliding of a block on an inclined plane. Numerical simulations of the Daguangbao landslide triggered by the Wenchuan earthquake were performed using the improved DDA method. The results illustrate that the shear strength degradation of discontinuities affect the evolution process, travel distance, and post-failure shape of the seismic landslide significantly.