Wave propagation in elasto-plastic granular systems

Due to the nonlinear nature of the inter-particle contact, granular chains made of elastic spheres are known to transmit solitary waves under impulse loading. However, the localized contact between spherical granules leads to stress concentration, resulting in plastic behavior even for small forces. In this work, we investigate the effects of plasticity in wave propagation in elasto-plastic granular systems. In the first part of this work, a force–displacement law between contacting elastic-perfectly plastic spheres is developed using a nonlinear finite element analysis. In the second part, this force–displacement law is used to simulate wave propagation in one-dimensional granular chains. In elasto-plastic chains, energy dissipation leads to the formation and merging of wave trains, which have characteristics very different from those of elastic chains. Scaling laws for peak force at each contact point along the chain, velocity of the leading wave, local contact and total dissipation are developed.     

The discrete element method with deformable particles

This work presents a new original formulation of the discrete element method (DEM) with deformable cylindrical particles. Uniform stress and strain fields are assumed to be induced in the particles under the action of contact forces. Particle deformation obtained by strain integration is taken into account in the evaluation of interparticle contact forces. The deformability of a particle yields a nonlocal contact model, it leads to the formation of new contacts, it changes the distribution of contact forces in the particle assembly, and it affects the macroscopic response of the particulate material. A numerical algorithm for the deformable DEM (DDEM) has been developed and implemented in the DEM program DEMPack. The new formulation implies only small modifications of the standard DEM algorithm. The DDEM algorithm has been verified on simple examples of an unconfined uniaxial compression of a rectangular specimen discretized with regularly spaced equal bonded particles and a square specimen represented with an irregular configuration of nonuniform‐sized bonded particles. The numerical results have been verified by a comparison with equivalent finite element method results and available analytical solutions. The micro‐macro relationships for elastic parameters have been obtained. The results have proved to have enhanced the modeling capabilities of the DDEM with respect to the standard DEM.     

Direct measurement of particle–particle interaction using micro particle interaction analyzer (MPIA)

Direct particle–particle contact force measurement was successfully conducted for realistic parameter determination to support discrete element method (DEM) simulation by using a newly developed force measurement of micro particle interaction analyzer (MPIA). In this system particle-to-particle distance and deformation can be controlled by nanometer accuracy. The system can be used for measuring not only short-distance deformation but also long-distance deformation that was validated by both elastic contact and liquid bridge interaction including rupture distance, respectively. Then, the system was applied to obtain plastic normal deformation characteristics such as coefficients of restitution of the spherical granules at low loading force less than 0.5 mN. Granules were prepared from two-stage pressure swing granulation (PSG) technique in a fluidized bed.     

Discrete element simulation of railway ballast: modelling cell pressure effects in triaxial tests

The paper investigates reproducing the effects of confining pressure on the behaviour of scaled railway ballast in triaxial tests in discrete element models (DEM). Previous DEM work, using a standard Hertzian elastic contact law with an elastic–perfectly plastic tangential slip model, has been unable to replicate the behaviour observed in laboratory tests across a range of confining pressures without altering both the material stiffness and the inter-particle friction. A new contact law modelling damage at the contacts between particles is introduced. Particle contact is via spherically-capped conical asperities, which reduce in height if over-stressed. This introduces plasticity to the behaviour normal to the contact surface. In addition, the inter-particle friction angle is varied as a function of normalized contact normal force. At relatively low normal forces the friction angle must be increased for peak mobilized friction angles to match the laboratory data, an effect that is attributed to interlocking at the scale of surface roughness. Simulation results show close agreement with laboratory data.     

塑性强化材料颗粒的微观接触摩擦研究

合理的颗粒接触摩擦模型是从微观角度研究各类结构面强度的基础,然而受制于岩土体颗粒在微观高应力环境中表现出的复杂接触特征,该问题一直未能很好解决。Fujimoto 在2000 年给出了受切向荷载作用的微凸体在理想弹性或完全塑性接触状态下的微观位移特性,但却缺少对塑性强化接触状态下的摩擦进行解析,难被用于分析微观高应力状态下岩土颗粒的接触摩擦。为此,以Fujimoto模型为基础,结合作者曾经提出的塑性强化接触变形理论,在构建塑性强化接触状态下颗粒微观位移模型的基础上,系统的研究了切向荷载作用下塑性强化材料颗粒的接触摩擦机理,阐明了不同接触状态下塑性强化材料颗粒的切向微观位移特征。最后通过算例分析显示了模型的合理性。结果表明:颗粒摩擦本质上是不同法向荷载不同接触状态区域按照不同摩擦类型提供摩擦的综合,而摩擦失稳就是接触面上微滑区扩大、粘着区缩小并消失的过程。     

基于弹-塑-断裂理论的镐型截齿截割机理研究与实验验证

为研究镐型截齿在截割煤岩过程中的截割机理,建立镐型截齿外轮廓的数学模型,基于弹性力学、塑性力学和断裂力学等理论,提出了煤岩截齿截割力作用下发生弹-塑-断裂失稳理论模型,以此来描述截齿截割煤岩机理,并依据理论模型分别推导出煤岩在弹性变形、塑性变形和断裂失稳状态下截齿截割力以及煤岩应力表达式。最后采用自主搭建实验设备进行不同截割倾角的镐型截齿截割煤岩实验,实验结果表明:煤岩在截齿截割力作用过程中,随着截齿截割深度的增加,截齿与煤岩的接触面积逐步增大,煤岩存在弹性变形阶段、塑性变形阶段以及断裂失稳阶段并与理论模型相符;截割倾角为90°、75°、60°时,截齿截割阻力理论值与实验值的均方根误差分别为0.082 kN、0.199 kN、0.204 kN,理论值与实验值相差较小,验证理论模型的正确性。     

Modeling progressive delamination of laminated composites by discrete element method

Discrete element method (DEM) was used to model progressive delamination of fiber reinforced composite laminates. The anisotropic composite plies were constructed through a hexagonal packing of particle elements. Contacts between the particles were represented by parallel bonds with the verified normal and shear elastic properties. The ply interface was characterized by a contact softening model with a bilinear elastic behavior which is similar to the cohesive zone model in the continuum mechanics. DCB, ELS and FRMM tests were simulated by the DEM model to assess its capability of modeling mode I, mode II and mix mode fracture of delamination, respectively. Good agreements were observed between the DEM and existing numerical and experimental results of loading curves, which confirmed that the DEM model can be used to simulate initiation and propagation of composite delamination, with more insights into microscopic material behavior, such as damage extension and plastic zone.     

CFD-DEM analysis of internal packing structure and pressure characteristics in compressible filter cakes using a novel elastic–plastic contact model

Cake-forming filtration is a proven method for separating particles from suspensions. Most filtration models are based on the simplification of incompressible and homogeneous cake structures. However, most filter cakes are in fact unevenly compressed by e.g. the high transmembrane pressures, leading to dense structures with high flow resistance at the filtration membrane. Experimental investigations of these inhomogeneous cakes are challenging due to mostly invasive procedures after filtration has already taken place. In contrast, numerical methods can provide extensive information about fluid flow, particle separation and cake formation during filtration. However, this requires that both elastic and plastic particle deformation and forces are modeled correctly. To achieve this, the present study implemented a novel elastic–plastic DEM model that only requires measurable material parameters and therefore does not need any fitting. Subsequently, previously measured material parameters for elastic–plastic cellulose-lactose pellets (MCC) were used to investigate the packing density, fluid pressure levels and contact forces inside compressible filter cakes using CFD-DEM coupling. A comparison with incompressible and elastically compressible filter cakes showed a significant difference in the filtration behavior. Due to plastic deformation, a strong increase of the packing density when nearing the filtration membrane was found, leading to higher flow resistance for the filtration process. For cyclic filtration events, only the plastically deformed cake showed reduced height recovery in a relaxed state.     

Experimental and numerical investigations of low velocity impact on functionally graded circular plates

This paper addresses low-velocity impact behaviour of functionally graded clamped circular plates. An experimental work was carried out to investigate the impact behaviour of FG circular plates which is composed of ceramic (SiC) and metal (Al) phases varying through the plate thickness by using a drop-weight impact test system. The influence of the compositional gradient exponent and impactor velocity on the contact forces and absorbed energies was concentrated on the tests. The explicit finite element method, in which a volume fraction based elastic–plastic model (the TTO model) was implemented for the functionally graded materials, was used to simulate their drop-weight impact tests. Effective material properties at any point inside FGM plates were determined using Mori–Tanaka scheme. The experimental and numerical results indicated that the compositional gradient exponent and impactor velocity more effective on the elasto-plastic response of the FG circular plates to a low-velocity impact loading. The comparison at the theoretical and experimental results showed that the use of the TTO model in modelling the elasto-plastic behaviour of FG circular plates results in increasing deviations between the numerical and experimental contact forces for ceramic-rich compositions whereas it becomes more successful for metal-rich compositions.     

A new soft-particle DEM model of micro-particle impact integrated adhesive,elastoplastic and microslip behaviors

Micro-particle impact is a problem of solid mechanics that is common in many applications. To address this problem, a new soft-particle DEM model of micro-particle impact is proposed, which incorporates adhesive, elastoplastic and microslip behaviors. The normal force model is developed as two contiguous loading stages: the elastic stage and the elastoplastic stage in which the transition is from the elastic deformation to fully plastic deformation. Most innovative in unloading, the normal force model is also evolved into two contiguous stages: unloading under elastic loading and unloading under elastoplastic loading in which it combines Hertz elastic model and Mesarovic-Johnson plastic model. The normal force model is further assumed as the one-way coupling with pressure-based Maw tangential model with the micro-slip behavior. Further model validations are performed by employing the experimental results in literatures. The validation results indicate that model predictions agree with the experimental data, and are demonstrated to be incredibly accurate than other models, particularly for restitution coefficients and critical sticking velocity. Furthermore we can find that the smaller size particle has a longer period of nonlinear loading, while the larger size particle has a longer period of linear loading. For tangential restitution coefficient at the small incident angle, a down trend may be due to the oscillation of the tangential force.     

Benchmark tests for verifying discrete element modelling codes at particle impact level

Over the past 30 years, the Discrete Element Method (DEM) has rapidly gained popularity as a tool for modelling the behaviour of granular assemblies and is being used extensively in both scientific and industrial applications. However, it is far from clear from reviewing the literature whether the large number of DEM codes have been verified and checked against fundamental benchmark problems. DEM simulates the dynamics of each particle in an assembly by calculating the acceleration resulting from all the contact forces and body forces. It is clearly necessary that such a model be validated or verified by comparing with experimental results, analytical solutions or other numerical results (e.g. Finite Element Analysis (FEA) results) at particle impact level. There appears to be no standard benchmark tests against which DEM codes can be verified. It is thus essential and useful to establish a set of standard benchmark tests to confirm that these DEM codes are modelling the particle dynamics as intended. This paper proposes a set of benchmark tests to verify DEM codes at particle impact level for spherical particles. The analytical solutions derived from elasticity theory for elastic normal collision of two spheres or a sphere with a rigid plane are first reviewed. These analytical solutions apply only to the elastic regime for normal impact. Secondly, the analytical solutions of frictional oblique impact between two spheres or a sphere with a rigid plane are scrutinized and derived. These analytical solutions originate from the dynamics principles and should be satisfied for any DEM contact force model with prescribed friction and restitution coefficients. A set of eight benchmark tests are designed and performed using commercial DEM codes. Test 1 and Test 2 consider the elastic normal impact of two spheres or a sphere with a rigid plane, whereas the other tests (Test 3–Test 8) investigate the energy dissipation due to the collision. These benchmark tests also involve different types of material. The DEM results were compared with the analytical solutions, experimental or FEA results found in the literature. All benchmark tests showed good to excellent match, providing a quantitative verification for the codes used in this study. These benchmark tests not only verify DEM codes but also enhance the understanding of fundamental impact phenomena for modelling a large number of particles.     

Nonlinear static/dynamic analysis for elasto-plastic laminated plates with interfacial damage evolution

A new analysis model, which includes the effects of interfacial damage, geometrical nonlinearity and material nonlinearity, is presented for elasto-plastic laminated plates. Based on the model, the nonlinear equilibrium differential equations for elasto-plastic laminated plates with interfacial damage are established. The finite difference method and iteration method are adopted to solve these equations. The nonlinear static and dynamic behaviors for the elasto-plastic laminated plates under the action of transverse loads are analyzed. Effects of interfacial damage on the stress and displacement distribution and nonlinear dynamic response are discussed in the numerical examples together with the comparison of nonlinear mechanical behaviors between the elastic and elasto-plastic laminated plates. Numerical results show that both the interfacial damage and plastic deformation put obvious influence on the mechanical properties of structures.     

单层网壳结构弹塑性屈曲分析的离散单元法研究

该文提出离散元塑性区法,即将任意2个球元的接触截面划分成若干小面积,通过各小面积的应力状态描述整个截面的塑性发展过程,较离散元塑性铰法更精确。该文推导了杆系离散元截面应变增量计算公式,建立了截面在三维应力-应变状态下的结构弹塑性本构方程、加卸载准则、截面内力积分公式以及计算分析流程。离散元弹塑性屈曲分析的追踪策略与弹性屈曲分析完全相同,即仍采用离散元力控制法或位移控制法。采用Fortran语言自编程序对若干单层网壳结构算例进行弹塑性屈曲分析,验证了离散元塑性区法的正确性和适用性,拓宽了离散单元法在工程领域的应用范围,为结构分析提供了新路径。     

On the Analysis of an EFG Method Under Large Deformations and Volumetric Locking

This work investigates a modified element-free Galerkin (MEFG) method when applied to large deformation processes. The proposed EFG method enables the direct imposition of the essential boundary conditions, as a result of the kronecker delta property of the special shape functions, constructed in the neighborhood of the essential boundary. The plasticity model assumes a multiplicative decomposition of the deformation gradient into an elastic and a plastic part and considers a J ₂ elasto-plastic constitutive relation that accounts for a nonlinear isotropic hardening. The constitutive model is written in terms of the rotated Kirchhoff stress and of the conjugate logarithmic strain measure. A total Lagrangian formulation is considered in order to improve the computational performance of the proposed algorithm. Here, aspects related to the volumetric locking are numerically investigated and an F-bar approach is considered. Some numerical results are presented, under axisymmetric and plane strain assumption, in order to attest the performance of the proposed method.     

Elasto-plastic analysis of a circular opening considering material dilatancy and elasto-plastic coupling effects

The deformation and fracture of a surrounding rock mass are important bases of underground engineering stability evaluation and bolting design. In this paper, an improved elastic-perfectly plastic-brittle model is employed to analyze the mechanical property weakening behavior of surrounding rock, including the elastic zone, plastic zone, and fracture zone. Examining a circular opening under hydrostatic pressure, material dilatancy, and elasto-plastic coupling were considered for an analytical study of stress and displacement of the surrounding rock. The influences of the supporting force, dilatancy, and elasto-plastic coupling were analyzed for a deep coal mine tunnel. It is shown that the supporting force cannot significantly change the state of the stress distribution, but it has an outstanding influence on the displacement of the surrounding rock. Dilatancy and elasto-plastic coupling can both cause a rapid increase of fractured zone deformation of the surrounding rock. Additionally, the majority of deformation within the surrounding rock was derived from expansion-related deformation of the fractured zone in a residual-strength state.     

Constitutive model and finite element formulation for large strain elasto-plastic analysis of shells

This contribution presents a refined constitutive and finite element formulation for arbitrary shell structures undergoing large elasto-plastic deformations. An elasto-plastic material model is developed by using the multiplicative decomposition of the deformation gradient and by considering isotropic as well as kinematic hardening phenomena in general form. A plastic anisotropy induced by kinematic hardening is taken into account by modifying the flow direction. The elastic part of deformations is considered by the neo-Hookean type of a material model able to deal with large strains. For an accurate prediction of complex through-thickness stress distributions a multi-layer shell kinematics is used built on the basis of a six-parametric shell theory capable to deal with large strains as well as finite rotations. To avoid membrane locking in bending dominated cases as well as volume locking caused by material incompressibility in the full plastic range the displacement based finite element formulation is improved by means of the enhanced assumed strain concept. The capability of the algorithms proposed is demonstrated by various numerical examples involving large elasto-plastic strains, finite rotations and complex through-thickness stress distributions.     

Fragments spawning and interaction models for DEM breakage simulation

Particles breakage occurs in many industrial applications. During the last decade many works have been devoted for modelling and simulating such processes. A new and innovative procedure of empirical comminution functions for Discrete Element Method (DEM) simulations (Kalman et al. in Granul Matter 11(4):253–266, 2009) posed the question how to introduce the fragments of the broken particle back into the computational domain. Daughter particles (Fragments) spawning and interaction imposes several problems during DEM simulation. Some of the main problems are: seeding (allocating) daughter particles and their initial conditions i.e. fragments locations, velocities and physical properties. This work focuses on the daughter particles seeding and the interaction between “sibling” particles for spherical particles. Fragments spawning and interaction algorithm for particle breakage during DEM simulation was developed. The algorithm enables prediction of particle comminution/attrition processes using DEM applications. The new algorithm can utilize any breakage function allowing unlimited fragment size fractions. In the proposed model, sibling particles can overlap without increasing the energy of the system in the simulation. Particle-particle and particle-wall interactions are calculated using the standard DEM calculations. Daughter particles interactions were calculated using the developed temporary contact radius model. The model was utilized to predict particle comminution in jet milling and particle attrition during pneumatic conveying with great successes.     

Critical time step for DEM simulations of dynamic systems using a Hertzian contact model

The discrete element method (DEM) typically uses an explicit numerical integration scheme to solve the equations of motion. However, like all explicit schemes, the scheme is only conditionally stable, with the stability determined by the size of the time step. Currently, there are no comprehensive techniques for estimating appropriate DEM time steps when a nonlinear contact interaction is used. It is common practice to apply a large factor of safety to these estimates to ensure stability, which unnecessarily increases the computational cost of these simulations. This work introduces an alternative framework for selecting a stable time step for nonlinear contact laws, specifically for the Hertz-Mindlin contact law. This approach uses the fact that the discretised equations of motion take the form of a nonlinear map and can be analysed as such. Using this framework, we analyse the effects of both system damping and the initial relative velocity of collision on the critical time step for a Hertz-Mindlin contact event between spherical particles.     

高层框架-斜交网格结构协同受力性能研究

高层框架-斜交网格结构是由高层斜交网格结构和框架结构组成的双抗侧力体系。采用理论推导和数值模拟的方法研究了该体系处于弹性和弹塑性状态下楼层水平剪力分配规律。针对高层斜交网格结构和框架结构协同工作变形特点,提出结构体系的弹性平面简化分析模型,进一步推导出结构侧向变形和剪力计算公式,并研究了结构处于弹性阶段时的剪力分配规律;采用非线性静力推覆法对结构体系在弹塑性阶段的变形特性、刚度退化和剪力分配规律进行了研究。结果表明:高层斜交网格结构和框架结构的协同工作性能良好,结构在弹性阶段,结构刚度的特征值及荷载分布对其剪力分配影响较大;结构进入塑性阶段,斜柱刚度退化速率较快,楼层剪力存在重分配现象。