Numerical study of inter‐particle bond failure by 3D discrete element method

The cohesive‐frictional nature of cementitious geomaterials raises great interest in the discrete element method (DEM) simulation of their mechanical behavior, where a proper bond failure criterion is usually required. In this paper, the failure of bond material between two spheres was investigated numerically using DEM that can easily reproduce the failure process of brittle material. In the DEM simulations, a bonded‐grain system (composed of two particles and bond material in between) was discretized as a cylindrical assembly of very fine particles connecting two large end spheres. Then, the bonded‐grain system was subjected to compression/tension, shear, rolling and torsion loadings and their combinations until overall failure (peak state) was reached. Bonded‐grain systems with various sizes were employed to investigate bond geometry effects. The numerical results show that the compression strength is highly affected by bond geometry, with the tensile strength being dependent to a lesser degree. The shear, rolling and torsion strengths are all normal force dependent; i.e., with an increase in the normal force, these strengths first increase at a declining rate and then start to decrease upon the normal force exceeding a critical value. The combined actions of shear force, rolling moment and torque lead to a spherical failure envelope in a normalized loading space. The fitted bond geometry factors and bond failure envelopes obtained numerically in this three‐dimensional study are qualitatively consistent with those in previous two‐dimensional experiments. The obtained bond failure criterion can be incorporated into a future bond contact model. Copyright © 2015 John Wiley & Sons, Ltd.     

A coupled 3‐dimensional bonded discrete element and lattice Boltzmann method for fluid‐solid coupling in cohesive geomaterials

This paper presents a 3D bonded discrete element and lattice Boltzmann method for resolving the fluid‐solid interaction involving complicated fluid‐particle coupling in geomaterials. In the coupled technique, the solid material is treated as an assembly of bonded and/or granular particles. A bond model accounting for strain softening in normal contact is incorporated into the discrete element method to simulate the mechanical behaviour of geomaterials, whilst the fluid flow is solved by the lattice Boltzmann method based on kinetic theory and statistical mechanics. To provide a bridge between theory and application, a 3D algorithm of immersed moving boundary scheme was proposed for resolving fluid‐particle interaction. To demonstrate the applicability and accuracy of this coupled method, a benchmark called quicksand, in which particles become fluidised under the driving of upward fluid flow, is first carried out. The critical hydraulic gradient obtained from the numerical results matches the theoretical value. Then, numerical investigation of the performance of granular filters generated according to the well‐acknowledged design criteria is given. It is found that the proposed 3D technique is promising, and the instantaneous migration of the protected soils can be readily observed. Numerical results prove that the filters which comply with the design criteria can effectively alleviate or eliminate the appearance of particle erosion in dams.     

DEM analyses of one-dimensional compression and collapse behaviour of unsaturated structural loess

Natural loess is a kind of under-consolidated and unsaturated loose granulates (silts) with its microstructure characterized with large voids and inter-particle cementation. This paper presents a distinct element method (DEM) to investigate its macro- and micro-mechanical behaviour (compression and collapse behaviour) under one-dimensional (1D) compression condition. A relationship between bond strength in DEM model and initial water content is used to develop a bond contact model for loess. Then, DEM structural loess samples are prepared by the multi-layer under-compaction method, and cemented with the bond contact model. The effect of water content and void ratio on compression and collapse behaviour of loess is numerically investigated by simulating 1D compression and wetting tests on the DEM material. The DEM results agree qualitatively with available experimental observations in literatures. The wetting-induced deformation is independent of the sequence of wetting and loading under 1D compression condition. The macroscopic yielding and collapse behaviours are associated with bond breakage on microscopic scale. Moreover, bonds break in one of the two failure types in the simulations, i.e. tensile failure and shear failure (compression-shear failure and tension-shear failure), with bonds broken firstly mainly due to tension followed by shear when the samples are compressed, while mainly due to shear when the samples are wetted under a certain pressure. In addition, the contact orientations and deviator fabrics of contacts under 1D compression and wetting were also investigated.     

A simple three‐dimensional distinct element modeling of the mechanical behavior of bonded sands

This paper presents a simple three‐dimensional (3D) Distinct Element Method (DEM) for numerical simulation of the mechanical behavior of bonded sands. First, a series of micro‐mechanical tests on a pair of aluminum rods glued together by cement with different bond sizes were performed to obtain the contact mechanical responses of ideally bonded granular material. Second, a 3D bond contact model, which takes into account the influences of bond sizes, was established by extending the obtained 2D experimental results to 3D case. Then, a DEM incorporating the new contact model was employed to perform a set of drained triaxial compression tests on the DEM bonded specimens with different cement contents under different confining pressures. Finally, the mechanical behavior of the bonded specimens was compared with the available experimental results. The results show that the DEM incorporating the simple 3D bond contact model is able to capture the main mechanical behavior of bonded sands. The bonded specimen with higher cement content under lower confining pressure exhibits more pronounced strain softening and shear dilatancy. The peak and residual strengths, the apparent cohesion and peak/residual friction angles, and the position and slope of the critical state line increase with increase in cement content. Microscopically, bond breakage starts when the system starts to dilate and the maximum rate of bond breakage coincides with the maximum rate of dilation. Bond breakage is primarily due to tension‐shear failure and the percentage of such failures is independent of both confining pressure and cement content. Copyright © 2015 John Wiley & Sons, Ltd.     

Size-dependent mechanical behavior of an intergranular bond revealed by an analytical model

The size of intergranular bonds significantly affects the macroscopic mechanical properties of geomaterials. A size-dependent bond contact model is desired in the distinct element method (DEM) for geomaterials formed by aggregates of bonded particles. This paper proposes an analytical solution of highly-precise stress fields of a biconcave bond between two identical disc-shaped particles under different loading paths based on Dvorkin’s solution. The Unified Strength theory is then introduced to obtain the initial failure domain in the bond. The proposed solution is consistent with results predicted by finite element simulations and experimental observations. The functions of bond stiffness with respect to all influencing parameters, i.e. bond width/thickness, particle radius and elastic parameters of bond material, are provided by the solution and empirically formulated by fitting a large number of analytical results. Additionally, the failure criterion or envelope under different combined loads is formulated for typical brittle bonds. The resulting failure criterion, approximated as an ellipsoid, depends on the size and material properties of the bonds. The proposed solution and equation can be implemented into a bond contact model used in DEM simulations of a geomaterial, where variation of bond sizes is significant and size-dependent contact model is important.     

胶结颗粒接触力学特性测试装置研制

为验证天然结构性砂土离散元模拟中接触模型及其参数的合理性,设计了一套用于理想胶结颗粒成型及实现不同加载条件下接触力学特性测试装置。通过胶结颗粒成型装置在两大小相同的铝棒间形成具有特定几何尺寸的胶结物,随后,采用一系列辅助加载装置实现简单及复杂加载条件下胶结颗粒接触力学特性的测试。试验结果表明：该装置可用于胶结颗粒在不同加载条件下接触力学特性的测试,实测胶结颗粒接触力学响应与天然砂土离散元中接触模型基本相符,且其抗剪和抗扭强度均随着法向压力的增大而增大,在三维应力空间中胶结颗粒强度包线呈椭圆抛物面状。     

A bond failure criterion for DEM simulations of cemented geomaterials considering variable bond thickness

A series of micromechanical tests were conducted to investigate the bond failure criterion of bonded granules considering the effect of bond thickness, with the aim of enhancing the bond contact model used in the distinct element simulations of cemented geomaterials. The granules were idealized in a two‐dimensional context as one pair of aluminum rods bonded by resin epoxy or cement. The mechanical responses of nearly 500 rod pairs were tested under different loading paths to attain the yield loads of bonded granules at variable bond thickness. This study leads to a generic bond failure criterion incorporating the effect of the bond thickness. The results show that the bond compressive resistance largely decreases with increasing bond thickness owing to the presence of the confinement at the bond‐particle interface. The strength envelopes obtained from the combined shear compression tests and combined torsion compression tests have identical functional form, and they decrease in size with increasing bond thickness but remain unchanged in shape. Given the same cementation material, the generic bond strength envelope in a three‐dimensional contact force space under different loading paths remains the same in shape but shrinks with the increase of bond thickness. Copyright © 2014 John Wiley & Sons, Ltd.     

考虑转动阻抗的粗粒土离散元模拟

转动阻抗被定义为作用颗粒接触上的一对对称力偶,用来抵抗颗粒之间的相互转动。将转动阻抗引入到离散元模拟中是对传统离散单元法的重要改进。开发出考虑颗粒转动阻抗的接触模型,并将其嵌入到PFC2D中,利用该模型进行粗粒土的双轴剪切数值模型试验,研究剪切过程中转动阻抗对粗粒土的宏细观力学性质的影响。结果显示,在宏观方面,颗粒转动阻抗对粗粒土的宏观力学行为（应力-应变及体应变-轴应变行为）有重要的影响,随着转动阻抗的增加,粗粒土的剪切强度和最大摩擦角随之增加,这与已有的研究结果一致,证明所建模型是可靠的;在微观方面,考察转动阻抗对粗粒土内部微观结构的影响发现,随着转动阻抗的增加,粗粒土的内部的接触数目减少,而粗粒土的剪切强度增加,表明转动阻抗能够提高粗粒土力链网络的稳定性,同时发现随着转动阻抗的增加,粗粒土的各向异性增加主要是强力链各向异性的增加,说明转动阻抗增强了强力链的传递力的能力以及抵抗力链屈曲破坏能力。数值模拟结果表明,增加颗粒转动阻抗,粗粒土出现组构与轴应变非共轴的现象。     

基于抗转模型的颗粒材料宏−细观关系研究

接触模型的宏?细观参数标定是成功使用离散元方法的关键。在离散元的接触模型中线性接触模型与抗转线性接触模型均可用于模拟砂性土的力学行为,其中抗转线性接触模型在模拟密砂的剪胀性方面具备优势。采用抗转线性接触模型对室内密实砂土三轴试验进行了离散元模拟,验证了抗转线性接触模型的可靠性;进而系统分析了颗粒间摩擦系数、刚度比和抗转动系数等细观参数与砂土峰值内摩擦角、残余内摩擦角、峰值剪胀角等宏观参数的相关关系并进行了验证;揭示了偏应力作用下,细观参数对密实砂土试样内部剪切带宽度与倾角变化的影响规律,提出了考虑剪胀角的剪切带倾角经验公式。通过研究建立了抗转线性接触模型宏?细观参数的量化关系并给出了标定参数的具体流程图,提出了快速标定宏观参数的方法并应用实例进行了验证,为采用抗转线性接触模型精准模拟密实砂土的力学特性提供依据。     

结构性黄土一维湿陷特性的离散元数值模拟

针对结构性湿陷性黄土大孔隙和胶结特性,应用离散元生成了不同含水率结构性黄土试样,研究试样的一维湿陷特性。首先,根据已有的结构性黄土试验资料和胶结颗粒材料离散元数值试验成果,建立胶结强度和初始饱和度之间的关系。其次,采用蒋明镜等提出的分层欠压法[1]和胶结模型[2]制得不同含水率结构性黄土离散元试样,然后进行不同含水率双线法和同一含水率4个压力下单线法湿陷试验的离散元数值模拟。数值模拟结果表明,提出的离散元分析方法能模拟天然结构性湿陷性黄土的主要力学性质,随着含水率的减少,结构屈服应力和最大湿陷压力增加,湿陷系数随着压力先增加后减小,湿陷起始压力为饱和试样的结构屈服应力,单线法湿陷后压缩曲线与饱和试样的压缩曲线接近。此外,模拟结果还表明,不同含水率结构性黄土离散元试样的最大湿陷系数与天然结构性湿陷性黄土相差较远,但在最大湿陷系数与孔隙比的比值上相接近;结构屈服对应着胶结的逐步破坏,湿陷伴随着大量的胶结破坏。提出了基于胶结点数目的损伤变量,研究了其在加载和湿陷过程中的变化规律。研究成果为认识黄土复杂力学特性和建立其本构理论提供了基础。     

On the linear elastic responses of the 2D bonded discrete element model

The bonded discrete element model (DEM) is a numerical tool that is becoming widely used when studying fracturing, fragmentation, and failure of solids in various disciplines. However, its abilities to solve elastic problems are usually overlooked. In this work, the main features of the 2D bonded DEM which influence Poisson's ratio and Young's modulus, and accuracy when solving elastic boundary value problems, are investigated. Outputs of numerical simulations using the 2D bonded DEM, the finite element method, a hyper elasticity analysis, and the distinct lattice spring model (DLSM) are compared in the investigation. It is shown that a shear interaction (local) factor and a geometric (global) factor are two essential elements for the 2D bonded DEM to reproduce a full range of Poisson's ratios. It is also found that the 2D bonded DEM might be unable to reproduce the correct displacements for elastic boundary value problems when the represented Poisson's ratio is close to 0.5 or the long-range interaction is considered. In addition, an analytical relationship between the shear stiffness ratio and the Poisson's ratio, derived from a hyper elasticity analysis and applicable to discontinuum-based models, provides good agreement with outputs from the 2D bonded DEM and DLSM. Finally, it is shown that the selection of elastic parameters used the 2D bonded DEM has a significant effect on fracturing and fragment patterns of solids.     

Investigation of influence of particle characteristics on the non‐coaxiality of anisotropic granular materials using DEM

As a result of deposition process and particle characteristics, granular materials can be inherently anisotropic. Many researchers have strongly suggested that the inherent anisotropy is the main reason for the deformation non‐coaxiality of granular materials. However, their relationships are not unanimous because of the limited understanding of the non‐coaxial micro‐mechanism. In this study, we investigated the influence of inherent anisotropy on the non‐coaxial angle using the discrete element method. Firstly, we developed a new discrete element method approach using rough elliptic particles and proposed a novel method to produce anisotropic specimens. Secondly, the effects of initial specimen density and particle characteristics, such as particle aspect ratio A _m, rolling resistance coefficient β , and bedding plane orientation δ , were examined by a series of biaxial tests and rotational principal axes tests. Findings from the numerical simulations are summarized as follows: (1) the peak internal friction angle ? _p and the non‐coaxial angle i both increase with the initial density, A _m and β , and they both increase initially and then decrease with δ in the range of 0–90°; (2) among the particle characteristics, the influence of A _m is the most significant; and (3) for anisotropic specimens, the non‐coaxial angle can be calculated using the double slip and rotation rate model. Then, an empirical formula was proposed based on the simulation results to depict the relationship between the non‐coaxial angle and the particle characteristics. Finally, the particle‐scale mechanism of non‐coaxiality for granular materials was discussed from the perspective of energy dissipation. Copyright © 2016 John Wiley & Sons, Ltd.     

Second‐order work analysis for granular materials using a multiscale approach

It has been established that the second‐order work criterion is a general necessary condition for all instabilities by divergence in rate‐independent granular materials. The relation between the values of discrete second‐order work at the intergranular contact point level and its global macroscopic value is recalled at the beginning of this paper. Then, the basic purpose of the paper is tackled by an analysis of the main features of second‐order work criterion in relation with the granular microstructure. For that, it is considered a novel micromechanical model (the so‐called ‘H‐microdirectional model’), which has the property to involve three scales: grain scale, mesoscale with a specific granular configuration and continuum mechanics macroscale. Eventually, these exhibited features (a bifurcation stress domain and some instability cones) are qualitatively compared with the ones provided by direct numerical simulations issued from a discrete element model. The ultimate goal is to analyse what happens at the granular scale, when the macrosecond‐order work is vanishing at the macrolevel. Copyright © 2013 John Wiley & Sons, Ltd.     

粒间胶结接触力学特性的三维试验研究

为实现结构性砂土离散元接触模型合理性的三维试验验证,设计了一套可用于三维半球形理想胶结颗粒成型及实现不同加载条件下的接触力学特性测试装置,制备了一定胶结尺寸的环氧树脂半球形颗粒胶结试样,在一系列辅助加载装置中初步开展了不同加载条件（拉伸、压缩、剪切、弯转、扭转）下的力学性能测试。结果表明,该装置可用于实现三维情况下胶结颗粒接触力学特性测试;不同加载条件下的实测试验结果与二维试验成果基本一致;峰值剪切、弯矩、扭矩随着法向荷载的增大呈现先增大后减小的趋势,存在一个相同的临界法向荷载。     

A micromechanical study of rolling and sliding contacts in assemblies of oval granules

The evolution of the microstructure of an assembly of cohesionless granular materials with associated pores, which carry the overall applied stresses through frictional contacts is a complex phenomenon. The macroscopic flow of such materials take place by the virtue of the relative rolling and sliding of the grains on the micro‐scale. A new discrete element method for biaxial compression simulations of random assemblies of oval particles with mixed sizes is introduced. During the course of deformation, the new positions of the grains are determined by employing the static equilibrium equations. A key aspect of the method is that, it is formulated for ellipse cross‐sectional particles, hence desirable inherent anisotropies are possible. A robust algorithm for the determination of the contact points between neighbouring grains is given. Employing the present methodology, many aspects of the behaviour of two‐dimensional assemblies of oval cross‐sectional rods have been successfully addressed. The effects of initial void ratio, interparticle friction angle, aspect ratio, and bedding angle on the rolling and sliding contacts are examined. The distribution of normals to the rolling and sliding contacts have different patterns and are concentrated along directions, which are approximately perpendicular to one another. On the other hand, the distribution of all contact normals (combined rolling and sliding) are close to that of rolling contacts, which confirm that rolling is the dominant mechanism. This phenomenon becomes more pronounced for higher intergranular friction angle. Characteristics of the rolling and sliding contacts are also discussed in the context of the force angle, which is the inclination of contact force with respect to the contact normal. Copyright © 2003 John Wiley & Sons, Ltd.     

Computational mechanics of the steel–concrete interface

Concrete cracking in reinforced concrete structures is governed by two mechanisms: the activation of bond forces at the steel–concrete interface and the bridge effects of the reinforcement crossing a macro‐crack. The computational modelling of these two mechanisms, acting at different scales, is the main objective of this paper. The starting point is the analysis of the micro‐mechanisms, leading to an appropriate choice of (measurable) state variables describing the energy state in the surface systems: on the one side the relative displacement between the steel and the concrete, modelling the bond activation; on the other hand, the crack opening governing the bridge effects. These displacement jumps are implemented in the constitutive model using thermodynamics of surfaces of discontinuity. On the computational side, the constitutive model is implemented in a discrete crack approach. A truss element with slip degrees of freedom is developed. This degree of freedom represents the relative displacement due to bond activation. In turn, the bridge effect is numerically taken into account by modifying the post‐cracking behaviour of the contact elements representing discrete concrete cracks crossed by a rebar. First simulation results obtained with this model show a good agreement in crack pattern and steel stress distribution with micro‐mechanical results and experimental results. Copyright © 2001 John Wiley & Sons, Ltd.     

Study of mechanical behavior and strain localization of methane hydrate bearing sediments with different saturations by a new DEM model

This paper presents a numerical investigation into mechanical behavior and strain localization in methane hydrate (MH) bearing sediments using the distinct element method (DEM). Based on the results of a series of laboratory tests on the bonded granules idealized by two glued aluminum rods and the available experimental data of methane hydrate samples, a pressure and temperature dependent bond contact model was proposed and implemented into a two-dimensional (2D) DEM code. This 2D DEM code was then used to numerically carry out a series of biaxial compression tests on the MH samples with different methane hydrate saturations, whose results were then compared with the experimental data obtained by Masui et al. [9]. In addition, stress, strain, void ratio and velocity fields, the distributions of bond breakage and averaged pure rotation rate (APR) as well as the evolution of strain localization were examined to investigate the relationships between micromechanical variables and macromechanical responses in the DEM MH samples. The numerical results show that: (1) the shear strength increases as methane hydrate saturation S_MH increases, which is in good agreement with the experimental observation; (2) the strain localization in all the DEM MH samples develops with onset of inhomogeneity of void ratio, velocity, strain, APR, and distortion of stress fields and contact force chains; and (3) the methane hydrate saturation affects the type of strain localization, with one shear band developed in the case of 40.9% and 67.8% methane saturation samples, and two shear bands formed for 50.1% methane saturation sample.     

Micromechanical analysis of the failure process of brittle rock

The failure process of brittle rock submitted to a compression state of stress with different confining pressures is investigated in this paper based on discrete element method (DEM) simulations. In the DEM model, the rock sample is represented by bonding rigid particles at their contact points. The numerical model is first calibrated by comparing the macroscopic response with the macroscopic response of Beishan granite obtained from laboratory tests. After the validation of numerical model in terms of macroscopic responses, the failure process of the DEM model under unconfined and confined compression is studied in micro‐scale in detail. The contact force network and its relation to the development of micro‐cracks and evolution of major fractures are studied. Confining pressure will prohibit the development of tensile cracks and hence alter the failure patterns. An in‐depth analysis of micro‐scale response is carried out, including the orientation distribution and probability density of stress acting on parallel bonds, the effect of particle size heterogeneity on bond breakage and the evolution of fabric tensor and coordination number of parallel bond. The proposed micromechanical analysis will allow us to extract innovative features emerged from the stresses and crack evolution in brittle rock failure process. Copyright © 2014 John Wiley & Sons, Ltd.     

Characterization of clastic rock using a biconcave bond model of DEM

This paper presents a biconcave bond model to investigate the effect of the cementation between grains on the mechanical behavior of rock. The proposed model considers the shape of the bonds among particles that have a biconcave cement form, based on observations of microscopic rock images. The general equations of the proposed model are based on Dvorkin theory. The accuracy and efficiency of the bond model is improved in three ways. After the biconcave bond model is implemented in the discrete element method software Particle Flow Code in 2 Dimensions, a series of numerical uniaxial compression tests were performed to investigate the relationships between the micro‐ to macro‐parameters. The simulations revealed that the biconcave bond model reflects the effect of micro‐parameters, such as the elastic modulus and Poisson's ratio of the cement, on the macroscopic deformation of cemented granular material. Variations in the bond geometry caused extremely diverse macro‐mechanical behaviors. Experimental results concerning rock demonstrate that the biconcave bond model accurately captures the mechanical behavior of intact rock and supports an innovative method for investigating the relationships between the micro‐ and macro‐parameters of cemented granular material. Copyright © 2016 John Wiley & Sons, Ltd.     

Modeling three‐dimensional hydraulic fracture propagation using virtual multidimensional internal bonds

Propagation of fractures, especially those emanating from wellbores and closed natural fractures, often involves Mode I and Mode II, and at times Mode III, posing significant challenges to its numerical simulation. When an embedded inclined fracture is subjected to compression, the fracture edge is constrained by the surrounding materials so that its true propagation pattern cannot be simulated by 2D models. In this article, a virtual multidimensional internal bond (VMIB) model is presented to simulate three‐dimensional (3D) fracture propagation. The VMIB model bridges the processes of macro fracture and micro bond rupture. The macro 3D constitutive relation in VMIB is derived from the 1D bond in the micro scale and is implemented in a 3D finite element method. To represent the contact and friction between fracture surfaces, a 3D element partition method is employed. The model is applied to simulate fracture propagation and coalescence in typical laboratory experiments and is used to analyze the propagation of an embedded fracture. Simulation results for single and multiple fractures illustrate 3D features of the tensile and compressive fracture propagation, especially the propagation of a Mode III fracture. The results match well with the experimental observation, suggesting that the presented method can capture the main features of 3D fracture propagation and coalescence. Moreover, by developing an algorithm for applying pressure on the fracture surfaces, propagation of a natural fracture is also simulated. The result illustrates an interesting and important phenomenon of Mode III fracture propagation, namely the fracture front segmentation. Copyright © 2012 John Wiley & Sons, Ltd.