Non-rigid disk-based DDA with a new contact model

In this paper, a new disk-based DDA formulation is presented. In the original disk-based DDA, disks are considered to be rigid and the penalty method is used to enforce disk contact constraints. In order to improve the accuracy of the disk-based DDA, new formulations of stiffness and force matrices for non-rigid disks using a new efficient contact model are presented in this paper. Blocks are considered deformable without need to do more computations for contact detection. In the proposed contact model, disk–disk and disk–boundary contacts are transformed into the form of point-to-line contacts and normal spring, shear spring and frictional force sub-matrices are derived by vector analysis. The penalty method is quite simple to implement, but has some major disadvantages. In the presented contact model, not only the simplicity of the penalty method is retained but also the limitations are overcome by using the augmented Lagrangian method. Moreover, unlike the contact model used in the original disk-based DDA, reference line can be obtained directly by using only coordinates of disk centers and their radii, and no more computations are needed. The validity and capability of the new disk-based DDA formulation are demonstrated by several illustrative examples.     

Dynamic sliding of tetrahedral wedge: The role of interface friction

The role of interface friction is studied by slow direct shear tests and rapid shaking table experiments in the context of dynamic slope stability analysis in three dimensions. We propose an analytical solution for dynamic, single and double face sliding and use it to validate 3D‐DDA. Single face results are compared with Newmark's solution and double face results are compared with shaking table experiments performed on a concrete tetrahedral wedge model, the interface friction of which is determined by constant velocity and velocity stepping, direct shear tests. A very good agreement between Newmark's method on one hand and our 3D analytical solution and 3D‐DDA on the other is observed for single plane sliding with 3D‐DDA exhibiting high sensitivity to the choice of numerical penalty value. The results of constant and variable velocity direct shear tests reveal that the tested concrete interface exhibits velocity weakening. This is confirmed by shaking table experiments where friction degradation upon multiple cycles of shaking culminated in wedge run out. The measured shaking table results are fitted with our 3D analytical solution to obtain a remarkable linear logarithmic relationship between friction coefficient and sliding velocity that remains valid for five orders of magnitude of sliding velocity. We conclude that the velocity‐dependent friction across rock discontinuities should be integrated into dynamic rock slope analysis to obtain realistic results when strong ground motions are considered. Copyright © 2011 John Wiley & Sons, Ltd.     

Applying disk-based discontinuous deformation analysis (DDA) to simulate Donghekou landslide triggered by the Wenchuan earthquake

In this paper, for the first time, disk-based discontinuous deformation analysis (DDA) is applied to simulate a real landslide triggered by an earthquake. For this purpose, the kinematic behaviour of the Donghekou landslide triggered by the Wenchuan earthquake is simulated and the results obtained using disk-based DDA are compared with actual data. The comparisons show that there is a good agreement between the results obtained using disk-based DDA and observed data. The simulation results provided an understanding of the failure behaviour and temporal evolution of the landslide. This study shows that disk-based DDA is a practical numerical tool that can be used to simulate the post-failure behaviour of landslides triggered by an earthquake.     

A detailed investigation of block dynamic sliding by the discontinuous deformation analysis

An extensive examination of the discontinuous deformation analysis (DDA) in block dynamic sliding modeling is carried out in this paper. Theoretical solutions for a single block sliding on an arbitrarily inclined plane by applying the horizontal/vertical seismic loadings to the sliding block as acceleration time histories or to the base as constraint displacement time histories are derived. As compared with the theoretical solutions, for a single block sliding, the DDA predicts the sliding displacements and block interaction forces accurately under various base incline angles and friction angles under both the harmonic loadings and a real seismic loading. The vertical seismic component may influence the block sliding displacements to different extent, and the DDA can capture these phenomena successfully and give accurate results. For the calculation of the single block relative sliding, both the theoretical and the DDA solutions indicate that applying the seismic accelerations as constraint displacement time histories (derived by integrating the seismic accelerations twice) to the base is equivalent to applying the seismic accelerations as volume forces to the sliding block in the opposite directions. The DDA modeling also demonstrates that this conclusion still stands for the case of multi‐block sliding. Copyright © 2012 John Wiley & Sons, Ltd.     

考虑滑床摩擦弱化的大光包滑坡运动机制DDA数值模拟研究

2008年汶川发生M_S8.0级地震,此次地震触发了大约20万处滑坡,其中大光包滑坡是汶川地震触发的最大规模的滑坡,其复杂的高速远程运动机理引起了国内外学者的广泛关注。本文结合热分解及动态结晶将地震中断层间摩擦弱化机制(闪速加热导致热分解及粉末润滑)运用于大光包滑坡高速远程运动的模拟。本文通过修改非连续变形分析(DDA)程序中强度参数的输入方式,以基于速度变化的强度参数取代原DDA程序中的常数强度参数,进而实现了摩擦系数随接触两侧相对速度变化的动态调整。运用修改后的DDA对大光包滑坡的运动过程进行模拟。模拟结果表明滑床摩擦弱化是导致大光包滑坡高速远程运动特征的重要原因,修改后的DDA由于考虑了滑床摩擦弱化能够更加合理地模拟滑坡的高速远程运动特征,本文模拟的大光包滑坡在地震作用下失稳后,由于滑床摩擦弱化,更多的能量转化为动能,高速滑体掠过黄洞子沟后,爬上对面的平梁子,最终由于平梁子的“急刹车”作用,滑体停止运动。与修改前DDA相比,修改后的DDA对大光包滑坡运动过程和最终堆积形态的模拟结果与已有文献记载和野外调查结果相吻合。这也间接证明了大光包滑坡滑动过程中由于白云岩间摩擦闪速加热导致热分解及粉末润滑造成的摩擦系数降低,可能是造成大光包滑坡高速远程运动的重要原因。     

地震作用下预应力锚索加固顺层岩坡极限分析

针对地震作用下预应力锚索加固顺层岩坡典型模型,提出了一种地震作用下预应力锚索加固顺层岩坡的极限分析改进方法,该方法考虑了边坡极限状态时锚索的受力变化。基于岩体刚性假设并考虑滑移过程锚索受力变化,结合极限平衡法推导出边坡加固预应力锚索的受力变化公式;地震作用仅考虑地震波传播至滑动面时产生的透射波对边坡稳定性的影响,从功率的角度出发,结合极限分析上限法与强度折减法,考虑预应力锚索受力变化,推导出地震作用下预应力锚索加固顺层岩坡的动安全系数计算理论公式;结合算例,采用考虑与不考虑滑移过程锚索受力变化两种计算方法,分析了在不同入射波波幅、入射角度及滑动面黏聚力、内摩擦角条件下两种方法计算结果的差异。算例结果表明：考虑与不考虑滑移过程锚索受力变化计算方法,计算得出的动安全系数变化规律一致,但考虑滑移过程锚索受力变化的改进计算方法得出的动安全系数动态变化幅度明显更小,反映了锚索的抗震作用效果,可为地震作用下该类型岩坡的预应力锚索加固设计分析提供参考。     

非连续变形分析中时间步及弹簧刚度取值研究

非连续变形分析（DDA）是一种隐式求解的动力学计算方法,且采用在块体界面加减刚硬弹簧的方式来满足块体界面无张拉和无嵌入的接触准则,其中时间步长和弹簧刚度两个物理量的取值直接影响DDA的计算结果。基于对DDA时间步和弹簧刚度在程序运行过程中的调整策略和块体接触的简化力学概念模型,研究了惯性力在DDA收敛求解中的作用过程。采用数值模拟试验对自由落体和斜面单滑块模型在3种力学状态下的相关力学问题进行了数值模拟研究,通过对自由落体运动的模拟,研究了时间步长单一因素对计算结果的影响规律,并初步确定了时间步长的合理取值区间。在此基础上,采用斜面单滑块模型,研究了时间步长和弹簧刚度对计算结果的共同影响,确定了不同时间步长条件下弹簧刚度的合理取值区间。研究成果表明,合适的时间步长和弹簧刚度的取值组合构成一个单连通参数取值域,当时间步和弹簧刚度的取值组合位于此“域”范围内时,DDA的计算结果是合理的。     

块系岩体滑移失稳中低摩擦效应的理论与试验研究

深部岩体具有块状层次结构,深部动载造成岩块发生相互间的振动脱离产生低摩擦效应,从而极易诱发原先处于平衡状态的岩体的动力变形破坏。在前人研究基础上,将块系岩体振动简化为等效质量-黏弹性模型,引入岩石摩擦滑移速率弱化模式,最终得到块系岩体滑移失稳计算模型。通过计算分析块系岩体自身特性及外荷载特性对岩块间低摩擦效应的影响。理论计算表明：水平静力及外扰动保持不变,增大岩块间弹性系数或者减小黏性系数,更容易引发岩体低摩擦滑移。随着冲击扰动、水平拉力幅值的增加,岩块的水平残余位移量值增加,当它们幅值超过一临界值时,岩块发生自持续滑移失稳运动。冲击扰动诱发岩块间不可逆位移、动力滑移失稳的临界能量与剪切力水平密切相关,在较大的剪切内力条件下,极其微弱的动力扰动即可诱发较大的岩块间不可逆位移甚至岩块的动力滑移失稳,随着剪切内力的减小,诱发岩块滑移失稳的能量阈值不断增大,当剪切内力低于岩块动摩擦强度时,单次冲击扰动只能诱发岩块间的不可逆位移。初步开展扰动诱发含初应力紫砂岩块体滑移试验,试验结果与理论计算基本符合,证明该模型的可行性。     

Gravitational instabilities triggered by fluid overpressure and downslope incision – Insights from analytical and analogue modelling

Fluid overpressure at the base of low-permeability strata reduces effective stress, allowing for gravitational sliding of the overlying cover. The force driving sliding is the slope-parallel component of the weight of the cover, whereas the resisting forces are the friction at the base of the cover and the buttressing resistance to shortening, which can be critically reduced by incision at the base of the slope. We developed an analytical model and undertook a series of analogue experiments to better understand the evolution of a sedimentary cover sliding above a low-permeability layer subjected to fluid overpressure. Where a downslope buttress was present, the sliding sheet length decreased with increasing pore-fluid pressure. In the absence of such buttress, the slide's length increased exponentially with increasing pore-fluid pressure. Another important difference dealt with geometry and kinematics. Buttressed slides consisted of one large slope-parallel mass rigidly translated and bounded by downslope thrusts and upslope normal faults. With increasing pore-fluid pressure, the contractional structures propagated upslope. By contrast, non-buttressed slides showed intense strain: deformation started with normal faults forming near the incision, then propagating upslope throughout the slide's evolution.     

一种改进步长自调的非连续变形分析法

非连续变形分析（DDA）是一种针对块体系统变形和位移求解的数值计算方法。引入Newmark方法于结构动力学微分方程中,考虑惯性力和阻尼力作用,改进时间步长自动调节,并实现DDA求解程序;比较研究Newmark方法中的线性加速法、常加速法和平均加速法在DDA程序中计算的收敛速度,讨论块体系统动力学计算过程中DDA方法对惯性力和阻尼力的添加和删除,并提出根据计算精度要求的误差控制实现方案。将改进的DDA方法模拟一个典型的煤与瓦斯突出过程,取得了满意的计算结果,该改进算法为DDA方法处理动力学问题提供新的途径     

Modelling dynamic deformation in natural rock slopes and underground openings with DDA: review of recent results

We present validations and applications of the numerical Discontinuous Deformation Analysis method (DDA) for different cases of dynamic loading in the context of rock mass deformation. Following a review of 2D and 3D-DDA validations against analytical solutions for single and double face sliding, we present dynamic DDA applications in natural rock slopes and underground openings. Modelling dynamic rock slope deformation is demonstrated using the case of Masada rock slopes, with some new findings on the dynamic deformation of overhanging cliffs in general. Modelling underground deformation is demonstrated using the case of an active open pit mine in Israel developed in a rock mass containing multiple karstic caverns. The DDA method is shown here to be a powerful numerical tool for modelling dynamic rock mass deformation when the interaction between multiple discrete elements dictates the expected global deformation.     

Boundary Element based Discontinuous Deformation Analysis

A Boundary Element based Discontinuous Deformation Analysis (BE‐DDA) method is developed by implementing the improved dual reciprocity boundary element method into the open close iterations based DDA. This newly developed BE‐DDA is capable of simulating both the deformation and movement of blocks in a blocky system. Based on geometry updating, it adopts an incremental dynamic formulation taking into consideration initial stresses and dealing with external concentrated and contact forces conveniently. The boundaries of each block in the discrete blocky system are discretized with boundary elements while the domain of each block is divided into internal cells only for the integration of the domain integral of the initial stress term. The contact forces among blocks are treated as concentrated forces and the open–close iterations are applied to ensure the computational accuracy of block interactions. In the current method, an implicit time integration scheme is adopted for numerical stability. Three examples are used to show the effectiveness of the algorithm in simulating block movement, sliding, deformation and interaction of blocks. At last, block toppling and tunnel stability examples are conducted to demonstrate that the BE‐DDA is applicable for simulation of blocky systems. Copyright © 2016 John Wiley & Sons, Ltd.     

Boundary setting method for the seismic dynamic response analysis of engineering rock mass structures using the discontinuous deformation analysis method

Large deformations and discontinuous problems can be calculated using the discontinuous deformation analysis (DDA) method by solving time steps, and this method is suitable for simulating the seismic dynamic response of engineering rock mass structures. However, the boundary setting must be carefully analyzed. In this paper, four boundary settings for the DDA method are investigated. First, the contributions to the DDA equations for nonreflecting boundaries (including the viscous boundary and the viscoelastic boundary) are deduced based on the Newmark method. Second, a free‐field boundary is introduced in the DDA method with boundary grid generation and coupling calculation algorithms to accurately simulate external source wave motion, such as earthquakes. Third, seismic input boundary treatments are intensively examined, and the force input method is introduced based on nonreflecting boundaries. Finally, the static‐dynamic unified boundary is implemented to ensure consistent boundary transformation. The boundary setting method in the DDA method is discussed, and the suggested treatments are used to analyze the seismic dynamic response of underground caverns. Copyright © 2015 John Wiley & Sons, Ltd.     

Computation of the safety factor for slope stability using discontinuous deformation analysis and the vector sum method

This study introduces the vector sum method into discontinuum-based methods by considering the sliding vector and the stress state of the discrete block system. The sliding direction computation and force projection in the new approach are detailed, and the safety factor is solved by explicit equations. The vector sum method is implemented in the discontinuous deformation analysis (DDA) program and is used to compute the safety factors for two numerical examples. A comparison of the solutions obtained with the theoretical analysis and limit equilibrium analysis demonstrates that the new method is suitable for calculating the safety factor of a slope.     

柔性支护黏性土基坑非极限被动土压力研究

经典的Rankine和Coulomb土压力计算理论均建立在土体达到极限平衡状态的基础上,并不适用于位移需要严格控制的基坑工程。以柔性支护的黏性土基坑边坡为研究对象,考虑边坡土拱效应、非极限状态下柔性支护结构与土体间内摩擦角以及黏聚力发挥值、土体内摩擦角以及黏聚力发挥值的影响,从黏性土应力莫尔圆出发,采用微层分析法建立静力平衡,搜索边坡土体潜在滑动面,推导柔性支护黏性土基坑的非极限被动土压力计算式。通过实例计算对比分析了本文计算理论与经典Rankine计算理论,推导公式计算得到的被动土压力小于Rankine计算值19%,合力作用位置低于Rankine计算值,作用位置距桩底距离较Rankine计算值小1.5%,计算得到的潜在滑动面为一水平倾角随深度逐渐减小的曲面,潜在滑动面范围小于Rankine极限状态滑动面。     

Nodal-based three-dimensional discontinuous deformation analysis (3-D DDA)

This paper presents a new numerical model that can add a finite element mesh into each block of the three-dimensional discontinuous deformation analysis (3-D DDA), originally developed by Gen-hua Shi. The main objectives of this research are to enhance DDA block’s deformability. Formulations of stiffness and force matrices in 3-D DDA with conventional Trilinear (8-node) and Serendipity (20-node) hexahedral isoparametric finite elements meshed block system due to elastic stress, initial stress, point load, body force, displacement constraints, inertia force, normal and shear contact forces are derived in detail for program coding. The program code for the Trilinear and Serendipity hexahedron elements have been developed, and it has been applied to some examples to show the advantages achieved when finite element is associated with 3-D DDA to handle problems under large displacements and deformations. Results calculated for the same models by use of the original 3-D DDA are far from the theoretical solutions while the results of new numerical model are quite good in agreement with theoretical solutions; however, for the Trilinear elements, more number of elements are needed.     

颗粒材料底面动摩擦系数特征研究

离散态颗粒物质具有明显不同于普通固体的界面摩擦特性,而摩擦系数是界面摩擦特性的主要表征参数之一。通过倾斜仪开展不同级配条件下颗粒材料的滑动摩擦试验,基于视频图像解析以及函数拟合方法,建立滑动位移与滑动时间的最佳函数拟合关系,分析滑动过程的加速度并推算底面动摩擦系数,研究颗粒粒径、质量配比等级配因素对颗粒材料底面动摩擦系数的影响。研究结果表明:(1)各级配颗粒材料的平均底面动摩擦系数随着运动时间的增加均呈线性减小趋势;(2)对于单粒径材料,与粗颗粒相比,细颗粒具有较大的底面动摩擦系数;(3)对于双粒径材料,随着细颗粒含量的增加,颗粒材料的平均底面动摩擦系数先急剧降低至最小值(细颗粒含量≤40%),后急剧增加(细颗粒含量40% ~60%),最终增加趋势明显变缓(细颗粒含量≥60%)。     

Implementation of displacement-dependent Barton-Bandis rock joint model into discontinuous deformation analysis

Discontinuous deformation analysis (DDA) has been widely applied in analyzing various rock engineering problems. As the joint strength play a vital role in the stability of jointed rock mass, this paper makes an attempt to implement the Barton-Bandis rock joint model into the DDA code to replace the original Mohr-Coulomb joint model. The developed Barton-Bandis joint model which is characterized by displacement-dependent shear strength is verified by experimental direct shear tests. An example of a block sliding on an inclined plane is used to demonstrate the capacity of the DDA-BB model in predicting the dynamic motion behavior of sliding blocks.     

基于固有振动频率的危岩安全监测试验研究

危岩体在经历地震和强降雨之后其抗滑力下降,从而导致重力作用失稳,是一种常见的破坏形式。危岩块体的失稳破坏取决于岩体结构面的状况,随着结构面的张开及黏结程度下降,危岩块体会发生失稳而崩塌,基于常规位移监测方法难以达到危岩块体监测预警的目的。而固有振动频率可以有效反映岩体的物理力学参数的变化,进而可以对其安全性做出预判。基于模型试验,在保持下滑力不变的情况下,通过固有振动频率来对滑坡内部的黏结力和摩擦力等力学指标进行分析。通过实际静摩擦力是否达到最大静摩擦力的方法来科学判识岩体的稳定情况。试验结果得出：计算的摩擦力可以有效分析岩体的安全性,并证明固有振动频率在降低至6 Hz以下,岩体开始趋于破坏。同时,得出由于摩擦力破坏后期占据抗滑力的比重逐渐升高,最高达到抗滑力的80%,从而造成破坏末期摩擦力不降反升的情况。基于固有振动频率的监测,可对岩体的损伤做出定量判断,也可评估岩体静摩擦力指标,从而实现扰动后岩体的安全评价。这种基于振动模态的岩体安全监测与损伤评价新的技术思路,将在实际工程中发挥重大作用。     

Seismic landslide simulations in discontinuous deformation analysis

Discontinuous deformation analysis (DDA) is a numerical approach used to simulate the post-failure behavior of a blocky assembly. Three available algorithms incorporate seismic impacts into DDA simulations for earthquake-induced slope failure. The following methods are used: directly applying time-dependent accelerations to falling/sliding blocks (Method 1); adding time-dependent accelerations to base block (Method 2); and time-dependently constraining seismic displacements of the base block (Method 3). However, incorrect absolute movements of falling/sliding and base blocks were obtained using Method 1. Additionally, relative movements between falling/sliding blocks and the base block are opposite to those simulated by the other two algorithms—Methods 2 and 3. Since locating an earthquake-induced landslide before an earthquake is extremely difficult, the seismic movements of base rock are recorded. Method 1 applies recorded seismic data to sliding blocks in conflict with d’Alembert’s principle of mechanics. Additionally, in Method 2, when the computation time step must be longer than the time in seismic data, computational results reveal abnormal base block displacements due to the non-zero velocity recorded at the end time of seismic data in seismic DDA. In this study, a novel algorithm to diminish the velocity of the base rock in the seismic analysis is utilized to modify Method 2. Furthermore, this work confirms that DDA with the modified Methods 2 and 3 is a practical approach for earthquake-induced landslide simulations.