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.

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.

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.

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.

A 3-D DDA damage analysis of brick masonry buildings under the impact of boulders in mountainous areas

In mountainous areas, geological disasters carrying large boulders can cause severe damage to the widely used masonry buildings due to the high impact forces. To better understand the damage of brick masonry buildings under the impact of boulders, a "block-joint" model is developed using threedimensional discontinuous deformation analysis(3-D DDA) to simulate the behaviour of the "brick-mortar" structure. The "block-joint" model is used to capture not only the large displacement and deformation of individual bricks but also the large-scale sliding and opening along the mortar between the bricks. The linear elastic constitutive model is applied to account for the non-plastic deformation behaviour of brick materials. Furthermore, the mechanical characteristics of the mortar are represented using the Mohr-Coulomb and Drucker-Prager criteria. To propose safe structural design schemes and effective reinforcement for brick masonry buildings, seven construction techniques are considered, includingdifferent grades of brick and mortar, effective shear areas and reinforced members. The proposed 3-D DDA model is used to analyse the velocity distribution and the key point displacements of the brick masonry building under the impact of boulders. The results show that upgrading the brick and mortar, increasing the wall thickness, making full use of the wall thickness, and adding a circular beam and structural column are very effective approaches for improving the impact resistance of brick masonry buildings.

三维DDA方法的验证及其在隧道围岩垮塌控制中的应用

为准确模拟隧道围岩垮塌灾害演化过程与锚杆加固处置,通过嵌入锚杆支护算法实现了三维非连续变形分析方法(DDA)中的锚杆支护模拟.首先,介绍了三维DDA方法模拟块体运动的基本原理,提出了三维DDA方法中锚杆支护算法;其次,针对隧道围岩垮塌灾害中的不同运动模式,采用三维DDA方法模拟了块体滑移、转动以及碰撞过程,验证了三维DDA方法模拟块体运动的准确性;最后,采用三维DDA方法模拟了工程尺度的节理岩体隧道围岩垮塌过程.结果表明:无支护条件下隧道洞周块体垮塌高度为3.9 m,洞周块体位移呈现出明显的各向异性特征;有支护条件下隧道洞周块体保持稳定状态,隧道左拱肩部位锚杆轴力较大,实际施工过程中可增加该区域锚杆长度.

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.

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.

Discontinuous deformation analysis based on complementary theory

The contact between blocks is treated by the open-close iteration in the conventional discontinuous deformation analysis (DDA),which needs to introduce spurious springs between two blocks in contact and to assume the normal stiffness and the tangential stiffness (the penalty factors). Unreasonable values of stiffness would result in numerical problems. To avoid the penalty factors and the open-close iteration,we reformulate the DDA as a mixed complementary problem (MiCP) and then choose the path Newton method (PNM) to solve the problem. Some examples including those originally designed by Shi are reanalyzed,which proves feasibility of the proposed procedure.

基于DDA方法的排碴井筒堵塞概率分析

依托某公路隧道的超大直径通风竖井工程,针对通风竖井施工中心小直径井筒排碴存在极大堵塞风险、严重影响建设工期的问题,考虑到井筒落石属于非连续变形动力学过程,因此,采用非连续变形分析(DDA)方法,通过对井筒落石进行数值建模,以球体作为基本块体单元,基于捆绑算法将球颗粒捆绑成相应形状块体,模拟块体运动过程中的碰撞、摩擦等,开展了针对性的小直径井筒落石堵塞概率分析。数值模拟结果表明,为避免小直径排碴井筒发生堵塞现象,应严格控制抛入排碴井筒的岩块尺寸与块度;钻爆法施工的小直径排碴井筒超欠挖现象越严重,井筒堵塞概率越大。模拟结果较为真实地反映了排碴井工况的工程实际。

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.

Some modifications to the process of discontinuous deformationanalysis

This paper presents a modified method of discontinuous deformation analysis(DDA). In the presented method, open-close iteration may not be needed, small penetration is permitted among blocks, and springs are added between contacting block pairs only when a penetration takes place. The three contact patterns(i.e. sliding, locking and opening) in original DDA method are not involved, and the recognition of these contact patterns and treatment of transformation among patterns are not required either,signi fi cantly saving the computing time. In a convex to concave contact, there are two candidate entrance edges which may cause uncertainty. In this case, we propose the angle bisector criterion to determine the entrance edge. The spring stiffness is much larger than Young's modulus in the original DDA, however we fi nd that the correct results can still be obtained when it is much smaller than Young's modulus. Finally,the penetrations by using penalty method and augmented Lagrangian method are compared. Penetration of the latter is 1/4 of the former. The range of spring stiffness for the latter is wider than the former,being 0.01-1 of the former. Both methods can lead to correct contact forces.

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.

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.

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.

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.

岩石圆盘径向压缩破坏的Voronoi子块体单元DDA方法模拟

非连续变形分析(DDA)方法是计算离散可变形块体系统力学行为的数值计算方法,可通过子块体单元DDA方法模拟岩石的开裂破坏。考虑到Voronoi多边形颗粒与细观尺度下岩石矿物晶粒形态的相似性,提出一种基于随机圆的Voronoi颗粒单元模型生成方法;并通过完整及带预制裂纹岩石圆盘径向压缩破坏的模拟,验证岩石破裂问题Voronoi子块体单元DDA模拟方法的适用性。结果表明,当子块体单元数较小时,圆盘表现出更高的整体强度;随着子块体单元数的增大,起裂处位置更接近真实,开裂破坏路径更清晰;子块体单元数较大时不同倾角预制裂纹圆盘破坏的模拟结果与实验结果高度吻合,并能有效反映圆盘中心加工小孔对开裂破坏路径的影响。使用Voronoi子块体单元DDA方法能够有效模拟岩石的开裂破坏过程,为进一步开展基于Voronoi颗粒单元模型的岩石开裂破坏模拟创造了条件。