Application of modified discontinuous deformation analysis to dynamic modelling of the Baige landslide in the Jinsha River

Accurate dynamic modeling of landslides could help understand the movement mechanisms and guide disaster mitigation and prevention.Discontinuous deformation analysis(DDA)is an effective approach for investigating landslides.However,DDA fails to accurately capture the degradation in shear strength of rock joints commonly observed in high-speed landslides.In this study,DDA is modified by incorporating simplified joint shear strength degradation.Based on the modified DDA,the kinematics of the Baige landslide that occurred along the Jinsha River in China on 10 October 2018 are reproduced.The violent starting velocity of the landslide is considered explicitly.Three cases with different violent starting velocities are investigated to show their effect on the landslide movement process.Subsequently,the landslide movement process and the final accumulation characteristics are analyzed from multiple perspectives.The results show that the violent starting velocity affects the landslide motion characteristics,which is found to be about 4 m/s in the Baige landslide.The movement process of the Baige landslide involves four stages:initiation,high-speed sliding,impact-climbing,low-speed motion and accumulation.The accumulation states of sliding masses in different zones are different,which essentially corresponds to reality.The research results suggest that the modified DDA is applicable to similar high-level rock landslides.

Particle Discontinuous Deformation Analysis of Static and Dynamic Crack Propagation in Brittle Material

Crack propagation in brittle material is not only crucial for structural safety evaluation,but also has a wideranging impact on material design,damage assessment,resource extraction,and scientific research.A thorough investigation into the behavior of crack propagation contributes to a better understanding and control of the properties of brittle materials,thereby enhancing the reliability and safety of both materials and structures.As an implicit discrete elementmethod,the Discontinuous Deformation Analysis(DDA)has gained significant attention for its developments and applications in recent years.Among these developments,the particle DDA equipped with the bonded particle model is a powerful tool for predicting the whole process of material from continuity to failure.The primary objective of this research is to develop and utilize the particle DDAtomodel and understand the complex behavior of cracks in brittle materials under both static and dynamic loadings.The particle DDA is applied to several classical crack propagation problems,including the crack branching,compact tensile test,Kalthoff impact experiment,and tensile test of a rectangular plate with a hole.The evolutions of cracks under various stress or geometrical conditions are carefully investigated.The simulated results are compared with the experiments and other numerical results.It is found that the crack propagation patterns,including crack branching and the formation of secondary cracks,can be well reproduced.The results show that the particle DDA is a qualified method for crack propagation problems,providing valuable insights into the fracture mechanism of brittle materials.

On the calibration and verification of Voronoi-based discontinuous deformation analysis for modeling rock fracture

Since its introduction,discontinuous deformation analysis(DDA)has been widely used in different areas of rock mechanics.By dividing large blocks into subblocks and introducing artificial joints,DDA can be applied to rock fracture simulation.However,parameter calibration,a fundamental issue in discontinuum methods,has not received enough attention in DDA.In this study,the parameter calibration of DDA for intact rock is carefully studied.To this end,a subblock DDA with Voronoi tessellation is presented first.Then,a modified contact constitutive law is introduced,in which the tensile and shear meso-strengths are modified to be independent of the bond lengths.This improvement can prevent the unjustified preferential failure of short edges.A method for imposing confining pressure is also introduced.Thereafter,sensitivity analysis is performed to investigate the influence of the calculated parameters and meso-parameters on the mechanical properties of modeled rock.Based on the sensitivity analysis,a unified calibration procedure is suggested for both cases with and without confining pressure.Finally,the calibration procedure is applied to two examples,including a biaxial compression test.The results show that the proposed Voronoi-based DDA can simulate rock fracture with and without confining pressure very well after careful parameter calibration.

FORMULATIONS OF THE THREE-DIMENSIONAL DISCONTINUOUS DEFORMATION ANALYSIS METHOD

This paper extends the original 2D discontinuous deformation analysis(DDA)method proposed by Shi to 3D cases,and presents the formulations of the 3D DDA.The formulations maintain the characteristics of the original 2D DDA approach.Contacts between the blocks are detected by using Common-Plane (C-P) approach and the non-smooth contact,such as of vertex-to-vertex,vertex- to-edge and edge-to-edge types,can be handled easily based on the C-P method.The matrices of equilibrium equations have been given in detail for programming purposes.The C program codes for the 3D DDA are developed.The ability and accuracy of the formulations and the program are verified by the analytical solutions of several dynamic examples.The robustness and versatility of the algorithms presented in this paper are demonstrated with the aid of an example of scattering of densely packed cubes.Finally,implications and future extensions are discussed.

Validation and application of three-dimensional discontinuous deformation analysis with tetrahedron finite element meshed block

In the last decade, three dimensional discontin- uous deformation analyses （3D DDA） has attracted more and more attention of researchers and geotechnical engineers worldwide. The original DDA formulation utilizes a linear displacement function to describe the block movement and deformation, which would cause block expansion under rigid body rotation and thus limit its capability to model block de- formation. In this paper, 3D DDA is coupled with tetrahe- dron finite elements to tackle these two problems. Tetrahe- dron is the simplest in the 3D domain and makes it easy to implement automatic discretization, even for complex topol- ogy shape. Furthermore, element faces will remain planar and element edges will remain straight after deformation for tetrahedron finite elements and polyhedral contact detection schemes can be used directly. The matrices of equilibrium equations for this coupled method are given in detail and an effective contact searching algorithm is suggested. Valida- tion is conducted by comparing the results of the proposed coupled method with that of physical model tests using one of the most common failure modes, i.e., wedge failure. Most of the failure modes predicted by the coupled method agree with the physical model results except for 4 cases out of the total 65 cases. Finally, a complex rockslide example demon- strates the robustness and versatility of the coupled method.

Dynamic process analysis of the Niumiangou landslide triggered by the Wenchuan earthquake using the finite difference method and a modified discontinuous deformation analysis

The Niumiangou landslide was the largest landslide triggered by the 2008 Wenchuan earthquake,which was significantly affected by the amplification effect of seismic acceleration.The ringshear experiments indicated that the materials in the source area of the Niumiangou landslide were subjected to friction degradation under a big shear displacement,which may result in rapid movement of the landslide.In order to better understand the landslide movement and study the effect of the friction degradation on movement mechanisms,the dynamic process of Niumiangou landslide was simulated with a new numerical method,which combines the finite difference method(FDM)and the discontinuous deformation analysis(DDA).First,the FDM was used to study the initiation time,amplification effect and velocity of the landslide.Afterwards,these initiation velocities were applied to the blocks in the DDA model by corresponding coordination in the FDM model.A displacementdependent friction model of the sliding surface was incorporated into DDA code to further understand the kinetic behavior of the landslide.The results show that the displacement-dependent friction strongly decreases the friction coefficient of sliding surface under a big displacement,which can obviously promote the run-out and velocity of landslide.The model output well matches the topographic map formed by the landslide.This implies that the proposed model can be applied to the simulation of earthquake-induced landslides with amplification effect,and the friction degradation model is important to clarify the movement mechanism of high-speed and long-distance landslides.

Discontinuous deformation analysis for subsidence of fractured formations under seepage: A case study

With the growing prominence of recycling projects of groundwater,the attention towards subsidence concerns in geological formations is intensifying.However,due to the long evolutionary time and complex underground discontinuities,the deformation field and subsidence mechanism are difficult to obtain.To this concern,this study implemented the Hydro-Mechanical Coupled Discontinuous Deformation Analysis(HM-DDA)in a groundwater recycling project located at a goaf mining site.The method for establishing numerical stratigraphic models and determining the required numerical parameters is introduced.This contributes to the comprehensive reconstruction of changes in in-situ stress within the goaf area,encompassing the initial stress equilibrium state,as well as the processes of water pumping and injection.The results indicated that the water injection process mitigated stress concentrations at both ends of the goaf area.Specifically,a 30-m rise in water head resulted in a corresponding elevation of the ground surface by 3.94 cm.

A continuous/discontinuous deformation analysis(CDDA)method based on deformable blocks for fracture modeling

In the framework of finite element meshes,a novel continuous/discontinuous deformation analysis(CDDA)method is proposed in this paper for modeling of crack problems.In the present CDDA,simple polynomial interpolations are defined at the deformable block elements,and a link element is employed to connect the adjacent block elements.The CDDA is particularly suitable for modeling the fracture propagation because the switch from continuous deformation analysis to discontinuous deformation analysis is natural and convenient without additional procedures.The SIFs(stress intensity factors)for various types of cracks,such as kinked cracks or curved cracks,can be easily computed in the CDDA by using the virtual crack extension technique(VCET).Both the formulation and implementation of the VCET in CDDA are simple and straightforward.Numerical examples indicate that the present CDDA can obtain high accuracy in SIF results with simple polynomial interpolations and insensitive to mesh sizes,and can automatically simulate the crack propagation without degrading accuracy.

Discontinuous deformation and displacement analysis: From continuous to discontinuous

A discontinuous deformation and displacement(DDD) analysis method is proposed for modelling the rock failure process. This method combines the rock failure process analysis(RFPA) method(based on finite element method) and discontinuous deformation analysis(DDA) method. RFPA is used to simulate crack initiation, propagation and coalescence processes of rock during the small deformation state. The DDA method is used to simulate the movement of blocks created by the multiple cracks modelled by the RFPA. The newly developed DDD method is particularly suitable for modelling both crack propagation and block movement during the rock failure process because of the natural and convenient coupling of continuous and discontinuous deformation analyses. The proposed method has been used to simulate crack initiation, propagation and coalescence within a slope as well as the block movement during the landslide process. Numerical modelling results indicate that the proposed DDD method can automatically simulate crack propagation and block movement during the rock failure process without degrading accuracy.

Elastoplastic model for discontinuous shear deformation of deep rock mass

Deep rock mass possesses some unusual properties due to high earth stress,which further result in new problems that have not been well understood and explained up to date.In order to investigate the deformation mechanism,the complete deformation process of deep rock mass,with a great emphasis on local shear deformation stage,was analyzed in detail.The quasi continuous shear deformation of the deep rock mass is described by a combination of smooth functions:the averaged distribution of the original deformation field,and the local discontinuities along the slip lines.Hence,an elasto-plastic model is established for the shear deformation process,in which the rotational displacement is taken into account as well as the translational component.Numerical analysis method was developed for case study.Deformation process of a tunnel under high earth stress was investigated for verification.

Condensed form of complementarity formulation for discontinuous deformation analysis

While the classical discontinuous deformation analysis(DDA) is applied to the analysis of a given block system, one must preset stiffness parameters for artificial springs to be fixed during the open-close iteration. To a great degree, success or failure in applying DDA to a practical problem is dependent on the spring stiffness parameters, which is believed to be the biggest obstacle to more extensive applications of DDA. In order to evade the introduction of the artificial springs, this study reformulates DDA as a mixed linear complementarity problem(MLCP) in the primal form. Then, from the fact that the block displacement vector of each block can be expressed in terms of the contact forces acting on the block, the condensed form of MLCP is derived, which is more efficient than the primal form. Some typical examples including those designed by the DDA inventor are reanalyzed, proving that the procedure is feasible.

Simulation of blast induced crater in jointed rock mass by discontinuous deformation analysis method

Rock blasting is a dynamic process accom panied with the propagations of shock waves and the dispersion of the explosion gas.This paper adopts the discontinuous deformation analysis(DDA)method to simulate the rock blasting process.A dynamic parameter adjustment and the non-reflecting boundary condition are implemented in the DDA method.The sub-block DDA method to simulate fracture problems is used.The blasting process in jointed rock mass is simulated by application of the explosion gas pressure on the expanding borehole walls and induced connected fracture surfaces around the boreholes.The blast craters with different overburdens are derived.The whole process including the explosion gas dispersion,borehole expansion,rock mass failure and cast,and the formation of the final blasting piles in rock blasting are well reproduced numerically.Parametric study for different overburdens is carried out,and the results are analyzed and discussed.

Parallelization of spherical discontinuous deformation analysis(SDDA) for geotechnical problems based on cloud computing environment

The computational efficiency of the traditional serial spherical discontinuous deformation analysis(SDDA) program has limited its application in geotechnical engineering problems that need a large number of spheres. The cloud computing technology is used to parallel the SDDA program for the first time in this research. The most computationally intensive portions of the SDDA program, i.e., contact detection and matrix solution, are parallelized with proposed algorithms. The accuracy of the cloud-based parallel SDDA program(CB-PSDDA) is verified first. Further efficiency tests show that significant speedup can be obtained with an 8-server configuration and the computing scale can be up to several tens of thousands of particles. The cloud-based parallelized SDDA program increases its potential in applications of deformation and failure analysis of large-scale and realistic geotechnical engineering problems.

Contact-Stress-Based Stress Recovery Methods for Discontinuous Deformation Analysis

Discontinuous deformation analysis(DDA)has been widely applied for the simulation of block systems that have many discontinuous surfaces.The penalty method is utilized to ensure that there are no penetrations between blocks.A linear polynomial function for displacement leads to a constant stress for a block,which cannot precisely describe the stress field within the block.Therefore,a high-order polynomial displacement function and a flue mesh are always used to improve the precision of the stress field.However,these means are not practical for simulating block systems that have many contacts.In this paper,the contact-stress-based stress recovery methods are proposed for DDA.High-precision solutions for the contact stresses on the boundaries of the blocks are utilized.The first-order Gaussian point of a block is the block's centroid,where the constant stress obtained via DDA is of higher precision.The high-precision solutions for the stresses are utilized in the least squares method to recover a single block's inner stress field.The proposed methods enhance the resolution of the stress field inside a single block without increasing the computational effort in the main iterative process for displacement in DDA.Numerical examples are simulated using both the finite element method(FEM)with a fine mesh and the proposed DDA program.The recovered DDA results can accurately describe the distribution of the stresses in a single block and,in some areas,have the same precision as the FEM results.Moreover,the precision of the proposed methods improves as the gradient of the contact stress on the boundary decreases.

A Coupling Model of the Discontinuous Deformation Analysis Method and the Finite Element Method

Neither the finite element method nor the discontinuous deformation analysis method can solve problems very well in rock mechanics and engineering due to their extreme complexities. A coupling method combining both of them should have wider applicability. Such a model coupling the discontinuous deforma- tion analysis method and the finite element method is proposed in this paper. In the model, so-called line blocks are introduced to deal with the interaction via the common interfacial boundary of the discontinuous deformation analysis domain with the finite element domain. The interfacial conditions during the incre- mental iteration process are satisfied by means of the line blocks. The requirement of gradual small dis- placements in each incremental step of this coupling method is met through a displacement control proce- dure. The model is simple in concept and is easy in numerical implementation. A numerical example is given. The displacement obtained by the coupling method agrees well with those obtained by the finite ele- ment method, which shows the rationality of this model and the validity of the implementation scheme.

NUMERIC ANALYSIS ON INFLUENCED FACTORS OF DISCONTINUOUS SURFACE DEFORMATION DUE TO STEEPLY-INCLINED SEAM MINING

This paper deals with the shape and influenced factors of surface non-continuous deformation due to mining. With finite element method, analysis are made to derive the relations between discontinuous deformation and mining affection, weak plane’s position & thickness, and mechanical property of weak-plane medium. The mutual affection of multiple weak-planes is also discussed. The results of the paper lay a foundation for constructing the calculation method of surface discontinuous deformation.

Surrounding rock deformation analysis of underground caverns with multi-body finite element method

Discontinuous deformation problems are common in rock engineering. Numerical analysis methods based on system models of the discrete body can better solve these problems. One of the most effective solutions is discontinuous deformation analysis （DDA） method, but the DDA method brings about rock embedding problems when it uses the strain assumption in elastic deformation and adopts virtual springs to simulate the contact problems. The multi-body finite element method （FEM） proposed in this paper can solve the problems of contact and deformation of blocks very well because it integrates the FEM and multi-body system dynamics theory. It is therefore a complete method for solving discontinuous deformation problems through balance equations of the contact surface and for simulating the displacement of whole blocks. In this study, this method was successfully used for deformation analysis of underground caverns in stratified rock. The simulation results indicate that the multi-body FEM can show contact forces and the stress states on contact surfaces better than DDA, and that the results calculated with the multi-body FEM are more consistent with engineering practice than those calculated with DDA method.

The Present Space-Time Motion and Deformation Features of the Northeastern Margin of the Qinghai-Xizang(Tibet) Block and Its Adjacent Area

On the basis of Discontinuous Deformation Analysis (DDA), and considering the moderate intrusion of specific block boundaries to different extents, the first-order block motion model is established for the northeastern margin of Qinghai-Xizang(Tibet) block and the kinematical model for depicting deformation of small regions as well by using GPS observations of three periods (1991, 1999 and 2001). By simulating, we obtained the motion features of the first-order blocks between the large WWN faults on the sides of the studied region, the distribution features of the principal strain rate field and the inhomogeneous motion features with space-time of the faults in the northern boundary of the Qinghai-Xizang (Tibet) block.

Simulation of failure process of jointed rock

A modified discontinuous deformation analysis (DDA) algorithm was proposed to simulate the failure behavior of jointed rock. In the proposed algorithm, by using the Monte-Carlo technique, random joint network was generated in the domain of interest. Based on the joint network, the triangular DDA block system was automatically generated by adopting the advanced front method. In the process of generating blocks, numerous artificial joints came into being, and once the stress states at some artificial joints satisfy the failure criterion given beforehand, artificial joints will turn into real joints. In this way, the whole fragmentation process of rock mass can be replicated. The algorithm logic was described in detail, and several numerical examples were carried out to obtain some insight into the failure behavior of rock mass containing random joints. From the numerical results, it can be found that the crack initiates from the crack tip, the growth direction of the crack depends upon the loading and constraint conditions, and the proposed method can reproduce some complicated phenomena in the whole process of rock failure.

Static and dynamic stability analysis using 3D-DDA with incision body scheme

Discontinuous deformation analysis （DDA） provides a powerful numerical tool for the analysis of discontinuous media. This method has been widely applied to the 2D analysis of discontinuous deformation. However, it is hindered from analyzing 3D rock engineering problems mainly due to the lack of reliable 3D contact detection algorithms for polyhedra. Contact detection is a key in 3-D DDA analysis. The limitations and advantages of existing contact detection schemes are discussed in this paper, and a new approach, called the incision body （IB）, is proposed, taking into account the advantages of the existing methods. A computer code 3DIB, which uses the IB scheme as a 3D contact detection algorithm, was programmed with Visual C^＋＋. Static and dynamic stability analysis for three realistic engineering problems has been carried out. Furthermore, the focus is on studying the stability of a gravity dam on jointed rock foundation and dynamic stability of a fractured gravity dam subject to earthquake shaking. The simulation results show that the program 3DIB and incision body scheme are capable of detecting 3D block contacts correctly and hence simulating the open-close and slide process of jointed block masses. In addition, the code 3DIB could provide an effective tool for evaluating the safety of 3D dam structures, which is quite important for engineering problems.