On the modeling of confined buckling of force chains

Buckling of force chains, laterally confined by weak network particles, has long been held as the underpinning mechanism for key instabilities arising in dense, cohesionless granular assemblies, e.g. shear banding and slip-stick phenomena. Despite the demonstrated significance of this mechanism from numerous experimental and discrete element studies, there is as yet no model for the confined buckling of force chains. We present herein the first structural mechanical model of this mechanism. Good agreement is found between model predictions and confined force chain buckling events in discrete element simulations. A complete parametric analysis is undertaken to determine the effect of various particle-scale properties on the stability and failure of force chains. Transparency across scales is achieved, as the mechanisms on the microscopic and mesoscopic domains, which drive well-known macroscopic trends in biaxial compression tests, are elucidated.     

颗粒材料破碎演化路径细观热力学机制

颗粒材料在高应力环境下会发生颗粒破碎现象,颗粒破碎不仅影响颗粒材料的力学特性,同时与大量工程问题密切相关.目前的相关研究主要集中在唯象地描述颗粒破碎的演化以及破碎对力学特性的影响层面,对颗粒破碎演化路径的物理机制研究较少.本文基于热力学框架,采用细观力学中细观-宏观的均匀化方法推导了颗粒体系弹性能和破碎能量耗散,并在最大能量耗散的假设下,在热力学框架内,建立了理想化的无摩擦球体颗粒等向压缩过程的弹性-破碎模型,阐述了颗粒材料破碎演化路径细观热力学机制.由于模型的推导不依赖任何唯象的经验公式,因此模型中包含的参数均有明确的物理意义.模型预测与前人试验结果对比表明,材料的初始级配对弹性压缩模量和破碎应力的影响并不相同:不同分形维数级配对应的弹性体变模量存在极大值,而破碎应力却随着分形维数的增大单调递增;颗粒破碎的演化符合最大能量耗散原理,且颗粒材料的压缩曲线可以分为弹性-破碎-拟弹性3个机制不同的阶段.      

Granular vortices: Identification,characterization and conditions for the localization of deformation

We relate the micromechanics of vortex evolution to that of force chain buckling and, on this basis, formulate the conditions for strain localization in a continuum model of dense granular media. Using the traditional bifurcation analysis of shear bands, we show that kinematic vortex fields are in fact solutions to the boundary value problem satisfying null boundary conditions. To establish an empirical basis for our study, we first develop a method to identify the location of the core and boundary of each vortex from a given displacement field in two dimensions. We then employ this method to characterize the residual deformation field (i.e., the deviation of particle motions from the continuum deformation) in a physical experiment and a discrete element simulation of dense granular samples submitted to biaxial compression. Vortices in the failure regime are essentially confined to the shear band. Primary vortices, the clear majority, rotate in the same direction as the shear band; secondary vortices, the so-called wakes, rotate in the opposite direction. Primary vortices align in spatial succession along the central axis of the band; wakes form next to the band boundaries, in between and beside two adjacent primary vortices. Force chain buckling, the governing mechanism for shear bands, is responsible for vortex formation in the failure regime. Vortex dynamics are consistent with stick-slip dynamics. From quiescent conditions of jamming or stick, vortical motions arise from force chain buckling and associated relative particle rotations and sliding; these in turn precipitate intermittent periods of unjamming or slip, evident in the attendant drops in stress ratio and bursts in both kinetic energy and local nonaffine deformation. A kinematic vortex field inside shear bands is proposed that is consistent with the equations of continuum mechanics and the underlying instability of force chain buckling: such a field is periodic with a repeating unit cell comprising a primary vortex at the center of the band, with two trailing wakes close next to the band boundaries.     

Modelling long-term deformation of granular soils incorporating the concept of fractional calculus

Many constitutive models exist to characterise the cyclic behaviour of granular soils but can only simulate deformations for very limited cycles. Fractional derivatives have been regarded as one potential instrument for modelling memory-dependent phenomena. In this paper, the physical connection between the fractional derivative order and the fractal dimension of granular soils is investigated in detail.Then a modified elasto-plastic constitutive model is proposed for evaluating the long-term deformation of granular soils under cyclic loading by incorporating the concept of factional calculus. To describe the flow direction of granular soils under cyclic loading, a cyclic flow potential considering particle breakage is used. Test results of several types of granular soils are used to validate the model performance.     

Structural stability and jamming of self-organized cluster conformations in dense granular materials

We examine emergent, self-organized particle cluster conformations in quasistatically deforming dense granular materials from the perspective of structural stability. A structural mechanics approach is employed, first, to devise a new stability measure for such conformations in equilibrium and, second, to use this measure to explore the evolving stability of jammed states of specific cluster conformations, i.e. particles forming force chains and minimal contact cycles. Knowledge gained on (a) the spatial and temporal evolution of stability of individual jammed conformations and (b) their relative stability levels, offer valuable clues on the rheology and, in particular, self-assembly of granular materials. This study is undertaken using data from assemblies of nonuniformly sized circular particles undergoing 2D deformation in two biaxial compression tests: a discrete element simulation of monotonic loading under constant confining pressure, and cyclic loading of a photoelastic disk assembly under constant volume. Our results suggest that the process of self-assembly in these systems is realized at multiple length scales, and that jammed force chains and minimal cycles form the basic building blocks of this process. In particular, 3-cycles are stabilizing agents that act as granular trusses to the load-bearing force chain columns. This co-evolutionary synergy between force chains and 3-cycles proved common to the different materials under different loading conditions. Indeed, the remarkable similarities in the evolution of stability, prevalence and persistence of minimal cycles and force chains in these systems suggest that these structures and their co-evolution together form a generic feature of dense granular systems under quasistatic loading.     

Topological evolution in dense granular materials: A complex networks perspective

One of the great challenges in the science of complex materials – materials capable of emergent behavior such as self-organized pattern formation – is deciphering their “inherent” structural design principles as they deform in response to external loads. We have been exploring the efficacy of techniques from complex networks to the study of dense granular materials as a means to: (i) uncover such design principles and (ii) identify suitable metrics that quantify the evolution of structure during deformation. Herein, we characterize the developing network structure and loss of connectivity in a quasistatically deforming granular medium from the perspective of complex networks. Attention is paid to the evolution of the contact and contact force networks at the local or mesoscopic level, i.e., a particle and its immediate neighbors, as well as the macroscopic level. We explore network motifs and other topological properties at these multiple length scales, in an attempt to find that which best correlates with the constitutive properties of nonaffine deformation and dissipation, spatially and with respect to strain. Key processes or rearrangement events that cause loss of connectivity within the material domain, e.g. microbanding and force chain buckling, are investigated. Network statistics of these processes, previously shown to be major sources of energy dissipation and nonaffine deformation, are then tied to corresponding trends observed in the evolving macroscopic network. It is shown that consideration of the unweighted contact network alone is insufficient to tie dissipation to loss of material connectivity.     

Spatial Behaviour in Pre-stressed Constrained Elastic Cylinders

Considered within the theory of small deformations superposed upon large is spatial behaviour for a prestressed laterally constrained elastic cylinder in equilibrium under zero body force and for sets of elastic coefficients that are not positive-definite. Certain cross-sectional integral measures are shown to be logarithmically convex implying at least exponentially increasing growth behaviour. For sufficiently long cylinders, this conclusion contradicts at most quadratic growth, and consequently the theory ceases to be valid indicating possible initiation of buckling and similar phenomena.     

Analysis of mesoscale effects in high-shear granulation through a computational fluid dynamics–population balance coupled compartment model

There is a need for mesoscale resolution and coupling between flow-field information and the evolution of particle properties in high-shear granulation. We have developed a modelling framework that compartmentalizes the high-shear granulation process based on relevant process parameters in time and space. The model comprises a coupled-flow-field and population-balance solver and is used to resolve and analyze the effects of mesoscales on the evolution of particle properties. A Diosna high-shear mixer was modelled with microcrystalline cellulose powder as the granulation material. An analysis of the flow-field solution and compartmentalization allows for a resolution of the stress and collision peak at the impeller blades. Different compartmentalizations showed the importance of resolving the impeller region, for aggregating systems and systems with breakage. An independent study investigated the time evolution of the flow field by changing the particle properties in three discrete steps that represent powder mixing, the initial granulation stage mixing and the late stage granular mixing. The results of the temporal resolution study show clear changes in collision behavior, especially from powder to granular mixing, which indicates the importance of resolving mesoscale phenomena in time and space.     

Micromechanical model for viscoelastic materials undergoing damage

We have derived a stress–strain relationship for viscoelastic materials undergoing damage using a granular micromechanics approach. This approach assumes the material to possess a granular meso-structure such that the material is treated as a discrete or a particulate system. By considering the particle kinematics in terms of Taylor series expansion, a continuum model of the discrete system is obtained. The material rate-dependence and damage are modeled by assuming appropriate inter-granular force–displacement relationships that satisfy thermodynamic constraints. The advantage of this micromechanical approach is that the resultant continuum model retains the discrete nature by incorporating the effect of nearest neighbor grain interactions through the inter-granular force–displacement relationship and orientation vector. As a result, the derived model has the ability to predict a number of material phenomena, such as loading-induced anisotropy, dilation or pressure sensitivity, and secondary creep, which often manifest due to material granularity.     

Selected local stability problems of channel section flanges made of aluminium alloys

The paper addresses the issue of local buckling of compressed flanges of cold-formed thin-walled channel columns and beams with nonstandard flanges composed of aluminium alloys. The material behaviour follows the Ramberg–Osgood law. It should be noted that the proposed solution may be also applied for other materials, for example: stainless steel, carbon steel. The paper is motivated by an increasing interest in nonstandard cold-formed section shaping in local buckling analysis problems. Furthermore, attention is paid to the impact of material characteristics on buckling stresses in a nonlinear domain. The objective of the paper is to propose a finite element method (FEM) model and a relevant numerical procedure in ABAQUS, complemented by an analytical one. It should be noted that the proposed FEM energetic technique makes it possible to compute accurately the critical buckling stresses. The suggested numerical method is intended to accurately follow the entire structural equilibrium path under an active load in elastic and inelastic ranges. The paper is also focused on correct modelling of interactions between sheets of cross section of a possible contact during buckling analysis. Furthermore, the FEM results are compared with the analytical solution. Numerical examples confirm the validity of the proposed FEM procedures and the closed-form analytical solutions. Finally, a brief research summary is presented and the results are discussed further on.     

可破碎颗粒体在动力载荷下的耗能特性

采用离散元的数值方法, 通过连接键将若干小颗粒绑定为一个具有不规则外形的大颗粒体, 设置不同连接键强度模拟了颗粒体在外加动力载荷下破碎过程, 并探讨其中系统能量耗散特性. 计算结果表明, 颗粒体的破碎程度决定了系统能量耗散率, 即内部耗能占外界输入能量的比例. 破碎率越高, 颗粒间相互摩擦和碰撞越剧烈,系统能量耗散率越高. 同时, 在循环载荷下系统内颗粒体破碎绝大部分发生在加载初期, 随着颗粒体的分解破碎速率逐渐减小, 系统耗能能力也随之降低.      

Buckling of yeast modeled as viscoelastic shells with transverse shearing

Yeast cells can be regarded as micron-sized and liquid-filled cylindrical shells. Owing to the rigid cell walls, yeast cells can bear compressive forces produced during the biotechnological process chain. However, when the compressive forces applied on the yeast go beyond a critical value, mechanical buckling will occur. Since the buckling of the yeast can change the networks in its cellular control, the experimental research of the buckling of the yeast has received considerable attention recently. In this paper, we apply a viscoelastic shell model to study the buckling of the yeast. Meanwhile, the turgor pressure in the yeast due to the internal liquid is taken into account as well. The governing equations are based on the first-order shear deformation theory. The critical axial compressive force in the phase space is obtained by the Laplace transformation, and the Bellman numerical inversion method is then applied to the analytical result to obtain the corresponding numerical results in the physical phase. The concepts of instantaneous critical buckling force, durable critical buckling force, and delay buckling are set up in this paper. And the effects of the transverse shear deformation and the turgor pressure on the buckling phenomena are also given. The numerical results show that the transverse shearing effect will decrease the instantaneous critical buckling force and the durable critical buckling force, while the turgor pressure will increase both of them.     

Hysteresis in shape-memory alloys

We present an overview of hysteresis phenomena in the martensitic transformation, and their relevance in the thermomechanical behaviour of shape-memory alloys. The first part of the paper introduces the concept of hysteresis, and the related phenomena of branching, dissipation and memory. The second part deals with revising some aspects of the thermomechanical behaviour of shape-memory alloys, emphasizing the hysteretic behaviour of single crystals and polycrystals under different driving conditions. The last part of the work is dedicated to the problem of modelling hysteresis phenomena in shape-memory alloys. Our focus is on phenomenological approaches which, as shown in the paper, account for the memory properties observed in hysteretic trajectories and open the possibility of deriving a generic energy balance for systems with hysteresis.     

Numerical study of coarse coal particle breakage in pneumatic conveying

Pneumatic conveying of coarse coal particles with various pipeline configurations and swirling intensities was investigated using a coupled computational fluid dynamics and discrete element method. A particle cluster agglomerated by the parallel-bond method was modeled to analyze the breakage of coarse coal particles. The numerical parameters, simulation conditions, and simulation results were experimentally validated. On analyzing total energy variation in the agglomerate during the breakage process, the results showed that downward fluctuation of the total particle energy was correlated with particle and wall collisions, and particle breakage showed a positive correlation with the energy difference. The correlation between the total energy variation of a particle cluster and particle breakage was also analyzed. Particle integrity presented a fluctuating upward trend with pipe bend radius and increased with swirling number for most bend radii. The degree of particle breakage differed with pipeline bending direction and swirling intensity: in a horizontal bend, the bend radius and swirling intensity dominated the total energy variations; these effects were not observed in a vertical bend. The total energy of the particle cluster exiting a bend was generally positively correlated with the bend radius for all conditions and was independent of bending direction.     

离散元法铁粉末压制中粒径分布对力链演化机制的影响

为阐明粒径分布对铁粉压制中体系内部细观力学行为的影响, 基于离散元理论, 通过改变铁粉颗粒粒径分布建立压制模型, 结合力链提取方法, 通过对力链空间分布、力链数目、力链长度和力链方向性的分析, 探究粒径分布对力链演化的影响机理. 研究结果表明: 不同粒径分布的粉体压制时形成的力链空间分布具有差异, 粒径分布范围越小, 形成的力链分布越集中, 反之, 粒径分布范围越大, 形成的力链分布越松散且均匀; 在粉末压制时, 粒径分布对力链数目也有影响, 具体表现为随着粉体的粒径分布范围变大, 力链总数逐渐减少; 粉体的粒径分布对颗粒形成短力链的数目起着显著影响, 而对力链长度的影响较为有限; 随着粒径分布范围的增大, 力链的方向由均匀分布逐渐集中在特定角度方向, 表现出一定各向异性, 形成的交叉力链网络结构有利于提高粉体致密化程度. 本文为从粉体粒径分布影响层面拓展粉末压制细观力学理论提供基础, 亦为进一步结合粉体粒径分布及体系内力链演变过程改善粉末致密化行为提供指导.      

Accelerating CFD–DEM simulation of processes with wide particle size distributions

Size-reduction systems have been extensively used in industry for many years. Nevertheless, reliable engineering tools to be used to predict the comminution of particles are scarce. Computational fluid dynamics (CFD)–discrete element model (DEM) numerical simulation may be used to predict such a complex phenomenon and therefore establish a proper design and optimization model for comminution systems. They may also be used to predict attrition in systems where particle attrition is significant. Therefore, empirical comminution functions (which are applicable for any attrition/comminution process), such as: strength distribution, selection, equivalence, breakage, and fatigue, have been integrated into the three-dimensional CFD–DEM simulation tool. The main drawback of such a design tool is the long computational time required owing to the large number of particles and the minute time-step required to maintain a steady solution while simulating the flow of particulate materials with very fine particles.The present study developed several methods to accelerate CFD–DEM simulations: reducing the number of operations carried out at the single-particle level, constructing a DEM grid detached from the CFD grid enabling a no binary search, generating a sub-grid within the DEM grid to enable a no binary search for fine particles, and increasing the computational time-step and eliminating the finest particles in the simulation while still tracking their contribution to the process.The total speedup of the simulation process without the elimination of the finest particles was a factor of about 17. The elimination of the finest particles gave additional speedup of a factor of at least 18. Therefore, the simulation of a grinding process can run at least 300 times faster than the conventional method in which a standard no binary search is employed and the smallest particles are tracked.     

DEM prediction of particle flows in grinding processes

Particle size reduction is a critical unit process in many industries including mineral processing, cement, food processing, pigments and industrial minerals and pharmaceuticals. The aim is to take large feed material and as efficiently as possible reduce the size of the particles to a target size range. Over time, a very large range of equipment has been developed to perform this for many materials and in many different conditions. Discrete element method (DEM) modelling is a computational tool that can allow detailed exploration of the particle flow and breakage processes within comminution equipment and can assist in developing a clearer and more comprehensive understanding of the detailed processes occurring within. In this paper, we examine the particle and energy flows in representative examples of the equipment used in many grinding processes. We study a 36′ semi‐autogenous mill used in primary grinding for mineral processing, a ball mill used for cement clinker grinding, a grinding table also used for cement grinding, a tower mill used for fine grinding both in mineral processing and for industrial minerals and finally for an Isamill, which is used for ultra‐fine grinding in mineral processing. Copyright © 2008 John Wiley & Sons, Ltd.     

Settling and spreading behaviour of particle clusters in quiescent liquids in confined vessels

Here we report experiments on particle cluster settling at high Reynolds number in quiescent liquid contained in a vessel. The particles were observed to spread at the vessel bottom surface in a near-circular annular shape after settling irrespective of the shape of the vessel cross-section and particle shape, size, and types. Effect of different parameters such as mass, type and aspect ratio of the particles, height, and viscosity of liquid was investigated on spreading behaviour. Formation of the hemispherical bottom cap of the cluster that bounces upon hitting the vessel bottom surface was found to be responsible for the final circular annular shape of the settled structure. Particle leakage from the cluster was seen in the form of a tail. In the liquid having viscosity beyond 100 cP, cluster breakage was observed that resulted in hindered settling and asymmetric shapes of finally settled particles. The observations are useful to understand the overall area over which settling and spreading of such clusters can be observed.     

Analysis of bending and buckling of pre-twisted beams: A bioinspired study

Twisting chirality is widely observed in artificial and natural materials and structures at different length scales. In this paper, we theoretically investigate the effect of twisting chiral morphology on the mechanical properties of elas- tic beams by using the Timoshenko beam model. Particular attention is paid to the transverse bending and axial buckling of a pre-twisted rectangular beam. The analytical solution is first derived for the deflection of a clamped-free beam under a uniformly or periodically distributed transverse force. The critical buckling condition of the beam subjected to its self- weight and an axial compressive force is further solved. The results show that the twisting morphology can significantly improve the resistance of beams to both transverse bending and axial buckling. This study helps understand some phenomena associated with twisting chirality in nature and provides inspirations for the design of novel devices and structures.     

On the determination of particle impact breakage in selection function

This paper presents a thorough study of particle impact breakage in selection function with a unified breakage criterion. The impact mode and breakage pattern for particulate materials are classified based on a significant review of well-established impact testers. It was found that the lack of a unified breakage criterion to determine the breakage probability disables a direct comparison of particle breakage propensity from different impact loading testers. The literature breakage models to describe the breakage probability are reviewed where the advantage and drawback of these models are scrutinized. The sourced literature breakage models are compared with the zeolite breakage datasets in a unified breakage criterion to evaluate the model performance. A novel computational modelling workflow for a milling process is proposed to provide a guidance in implementing the digital twin in milling process prediction. The breakage probability models, i.e. the selection functions are comprehensively assessed in population balance model to examine the model serviceability. The model simplicity and fidelity in the model assessment are specifically discussed and the value of digital twin in substantially reducing the experimental trials is highlighted.