Application of the Hertz formulation in the discrete element model of pressure-assisted sintering

This paper presents the numerical modelling of initial powder compaction and pressure-assisted sintering performed by original viscoelastic discrete element model. The research is focused on the influence of the type of the model representing an elastic part of interparticle force. Two elastic contact models—linear and nonlinear Hertz model—have been implemented and used to analyse interaction of NiAl powder particles during compaction and sintering process. Numerical models have been validated using own experimental results. Microscopic effects (particle penetration) and macroscopic changes (relative density) have been compared. It has been shown that although both models represent properly macroscopic behaviour of the material at the sintering process, the Hertz model produces the results closer to the real experimental ones during the initial compaction stage. Evaluation of macroscopic quantities enables implementation of the discrete element model in the framework of the multiscale modelling framework which is currently developed for sintering processes.     

Microstructure evolution and densification behavior of TiC/316L composite powders during cold/warm die compaction and solid-state sintering: 3D particulate scale numerical modelling and experimental validation

To identify the microstructure evolution and densification behavior of TiC/316L composites in powder metallurgy (PM) process, 3D particulate scale numerical simulations were conducted to reproduce the cold/warm compaction and solid-state sintering of TiC/316L composite powders with corresponding physical experiments being carried out for model validation. The effects of compaction parameters and sintering temperature on the densification behavior of TiC/316L composite powders were systemically investigated. The particle deformation and morphology, stress/strain and microstructure evolutions, and grain size distribution in the whole process were characterized and compared to further illustrate the densification behavior and the underlying dynamics/mechanisms. The results show that compared with the cold compaction, the warm compaction can not only achieve higher relative density, smaller and more uniform equivalent stress, and weaker spring back effect, but also improve the friction condition among powder particles. The plastic deformation of 316L particles is the main densification mechanism during compaction. In the solid-state sintering of TiC/316L compacts, the densification is mainly indicated by shrinkage and vanishing of large residual pores along with the growth of the sintering necks, accompanied by the particle movement and growth along the boundary regions. Meanwhile, the particle displacement and grain size distribution are more uniform in the warm compacted TiC/316L component. Moreover, the equivalent (von Mises) stress in 316L particles is smaller than that in TiC particles.     

Size Ratio Effects on Particle Contact Evolution at Uniaxial Powder Compaction

Cold compaction of composite powders has been analyzed using a discrete element method (DEM). Powder aggregates consisting of up to approximately 10,000 particles and formed by two powder populations with known material strength and size ratios have been compacted both isostatically and uniaxially (die compaction). The particles were assumed constitutively to be perfectly plastic or rigid and as a result, local contacts between the particles were described by a linear force-displacement relation given by previous in-depth analyses of spherical indentation problems. Particular emphasis has been placed on investigating the particle contact evolution at die compaction and to compare the results with previous ones pertinent to the isostatic case. Consequently, the predictive capability of the fundamental assumptions frequently used in theoretical analyses of compaction problems is determined for a uniaxial situation. The main conclusion is that size ratio effects are substantial at die compaction and when such features are present, theoretical predictions overestimates the (average) number of contacts per particle. It was also found that the mechanical behaviors at isostatic and die compaction are very similar even though die compaction values are slightly higher at high values on the relative density of powder materials.     

Approach the powder contact force,voidage, tensile stress,wall frictional stress and state diagram of powder bed by simple pressure drop monitoring

The evaluation of the mechanical properties and the state of a powder bed are essential for industrial powder operations. We assume that the bed incipient yield is approximately the bed incipient fluidization, and the particle contact force, the bed voidage, the bed tensile stress and the bed-wall frictional stress can be determined by simple pressure drop monitoring when gradually increasing the superficial gas velocity from zero. A two-dimensional powder bed voidage-tensile stress state diagram at zero shear stress under anisotropic consolidation is initially prepared. For the sample powder bed, we show that the isotropic tensile stress estimated by the powder yield locus extrapolation, 340 Pa−770 Pa, from a shear tester is different from the anisotropic tensile stress evaluated, 120 Pa–180 Pa, by the pressure drop overshoot approximation.     

Spherical indentation of a transversely isotropic elastic half-space reinforced with a thin layer

The frictionless spherical indentation test is considered for a transversely isotropic elastic half-space reinforced with a thin layer whose flexural stiffness is negligible compared to its tensile stiffness. It is assumed that the deformation of the reinforcing layer can be treated as the generalized plane stress state. Closed-form analytical approximate equations for the maximum contact pressure, contact radius, and contact force are presented. The isotropic case is considered in detail.     

Mechanics of powders in the initial rearrangement stage of liquid-phase sintering

The mechanical response of powders in the initial rearrangement stage of liquid-phase sintering (LPS), where capillary force is the driving force for densification, is discussed in this paper. The model which is described in detail elsewhere [Xu, K., Mehrabadi, M.M., 1997, A micromechanical model for the initial rearrangement stage of liquid-phase sintering, Mech. Mater. 25, 137–157.] can be used to determine the dependence of the overall volume change on such factors as local liquid volume, uniformity of liquid distribution, contact angle, initial particle distance, initial confining pressure, particle size and viscosity. Here, the discussion is limited to the influence of macroscopic liquid volume ratio on the volume change behavior of the powder system. The results obtained are in qualitative agreement with the observation.     

Sintering of Nano-Particle Powders: Simulations and Experiments

The sintering and consolidation of nano-particle powders were investigated using both computer simulation and direct experimentation. Molecular dynamics was employed to investigate the sintering of Cu nano-particles. Both spherical and cylindrical particles were employed in these simulations. The work demonstrates that sintering of these systems can take place on a time scale of tens of picoseconds, owing to the high shear stresses that develop at small particle contacts. For two and three particle sintering, it was observed that mass transport occurs by plastic flow. It was also found that crystallographically misaligned particles rotate to form low energy boundaries. The influence of the initial packing configuration was also investigated. Full densification occurs during pressureless sintering at room temperature only if the nano-particle assembly is close-packed. The effect of both isostauc and uniaxial stress on densification of Cu was also studied. Grain boundary sliding and grain boundary relaxation were also investigated.

Experimental measurements of densification of bulk specimens undergoing shear deformation provide information on the macroscopic sintering behavior of large assemblies of nano-crystals. The constitutive law for densification of nano-crystalline TiO₂ has been obtained. Attempts are made to relate the observed macroscopic sintering behavior of this system to the microscopic processes elucidated by the simulations.     

Plasticity and mass-transfer in contacting nanoparticles

The atomic structure of interparticle contact interfaces which present a new type of interface, is reviewed. Such interfaces show long-range elastic stress fields and a marked tendency to reconstruct themselves into more equilibrium structures similar to conventional grain boundaries. The methods of reconstruction (ageing) of the interfaces are analysed in detail. The elastically stressed state in the volume of nanoparticles which have few contacts with neighbouring particles in the ensemble, is investigated. Plasticity and mass-transfer processes leading to the relaxation of these contact stresses are considered. The mechanisms of generation of dislocations and critical pressures of compaction are discussed in detail. Stress-stimulated diffusion initiated by contact pressures is compared with the traditional mass-transfer caused by surface tension during the sintering of conventional coarse-grained powders.     

A linear model of elasto-plastic and adhesive contact deformation

Rigorous non-linear models of elasto-plastic contact deformation are time-consuming in numerical calculations for the distinct element method (DEM) and quite often unnecessary to represent the actual contact deformation of common particulate systems. In this work a simple linear elasto-plastic and adhesive contact model for spherical particles is proposed. Plastic deformation of contacts during loading and elastic unloading, accompanied by adhesion are considered, for which the pull-off force increases with plastic deformation. Considering the collision of a spherical cohesive body with a rigid flat target, the critical sticking velocity and coefficient of restitution in the proposed model are found to be very similar to those of Thornton and Ning’s model. Sensitivity analyses of the model parameters such as plastic, elastic, plastic-adhesive stiffnesses and pull-off force on work of compaction are carried out. It is found that by increasing the ratio of elastic to plastic stiffness, the plastic component of the total work increases and the elastic component decreases. By increasing the interface energy, the plastic work increases, but the elastic work does not change. The model can be used to efficiently represent the force-displacement of a wide range of particles, thus enabling fast numerical simulations of particle assemblies by the DEM.     

A combined finite‐discrete element method for simulating pharmaceutical powder tableting

The pharmaceutical powder and tableting process is simulated using a combined finite‐discrete element method and contact dynamics for irregular‐shaped particles. The particle‐scale formulation and two‐stage contact detection algorithm which has been developed for the proposed method enhances the overall calculation efficiency for particle interaction characteristics. The irregular particle shapes and random sizes are represented as a pseudo‐particle assembly having a scaled up geometry but based on the variations of real powder particles. Our simulations show that particle size, shapes and material properties have a significant influence on the behaviour of compaction and deformation. Copyright © 2005 John Wiley & Sons, Ltd.     

Approach to structural anisotropy in compacted cohesive powder

We investigate the mesoscopic regime between microscopic particle properties and macroscopic bulk behavior and present a complementary approach of physical experiments and discrete element method simulations to explore the development of the microstructure of cohesive powders during compaction. On the experimental side, a precise micro shear tester \((\mu \hbox {ST})\) for very small powder samples has been developed and integrated into a high resolution X-ray microtomography (XMT) system. The combination of \(\mu \hbox {ST}\) and XMT provides the unique possibility to access the 3D microstructure and the particle network inside manipulated powder samples experimentally. In simulations we explore the structural changes resulting from compaction: a Hertzian contact model is utilized for compaction of an isotropic initial configuration created by a geometrical algorithm. As a first result of this approach we present the analysis of the compaction of slightly cohesive \(\hbox {SiO}_2\) particles with special regard to bulk density, heterogeneity, compaction law and structural anisotropy.     

Spark plasma sintering of ductile ceramic particles: study of LiF

Densification of cuboidal micrometer-sized lithium fluoride particles as ductile ceramic by spark plasma sintering (SPS) was investigated. Specimens were fabricated at different pressures and temperature conditions, ranging from 2 to 100 MPa at 500 °C and from 200 to 700 °C under 100 MPa of applied pressure, respectively. Dense specimens of 99 % relative density were fabricated by heating to 500 °C under constant pressure of 100 MPa. The densification showed first compaction by particle rearrangement, followed by plastic deformation via dislocation glide. Hot-pressing models were used to describe the densification by considering the temperature dependences of the yield stress, the strain hardening behavior and coefficients, and the pore size and shape dependences on the applied stress. A good agreement was found between the experimental and the theoretical densification curves. At low pressure of 2 MPa, the densification occurs by particle sliding, assisted by viscous flow at their surfaces, and local plastic deformation at the particle contacts, due to the intensified local stress. Finally, the micrometer-sized structural features and the contiguity achieved by plastic deformation at the start of spark plasma sintering (SPS) nullify any field effects in this model system at higher pressures; good agreement was obtained with expected conventional hot pressing.     

A New Algorithm for Simulating the Random Packing of Monosized Powder in CIP Processes

A new model for monosized particle packing is developed in this study in simulating CIP powder compaction. The model uses a central growth method by which particles are placed from a central particle outwards until a specified container is filled. The simulation program is able to generate a particle packing of 5000 particles with characteristics that are similar to experimental results with significantly short computational time. A particle packing with a high occurrence of contacts among particles is effectively simulated using this model. It is shown that this work is an improvement in predicting the coordination number of the compact compared with the predictions of previous models. The average coordination number predicted by this model is 7·0 compared to 6·0-6·5 obtained by other simulations.     

Moving finite-element mesh model for aiding spark plasma sintering in current control mode of pure ultrafine WC powder

The intricate bulk and contact multiphysics of spark plasma sintering (SPS) together with the involved non-linear materials’ response make the process optimization very difficult both experimentally and computationally. The present work proposes an integrated experimental/numerical methodology, which simultaneously permits the developed SPS model to be reliably tested against experiments and to self-consistently estimate the overall set of unknown SPS contact resistances. Unique features of the proposed methodology are: (a) simulations and experiments are conducted in current control mode (SPS-CCm); (b) the SPS model couples electrothermal and displacement fields; (c) the contact multiphysics at the sliding punch/die interface is modeled during powder sintering using a moving mesh/moving boundary technique; (d) calibration and validation procedures employ both graphite compact and conductive WC powder samples. The unknown contact resistances are estimated iteratively by minimizing the deviation between predictions and on-line measurements (i.e., voltage, die surface temperature, and punch displacement) for three imposed currents (i.e., 1,900, 2,100, 2,700 A) and 20 MPa applied pressure. An excellent agreement is found between model predictions and measurements. The results show that the SPS bulk and contact multiphysics can be accurately reproduced during densification of ultrafine binderless WC powder. The results can be used to benchmark contact resistances in SPS systems applicable to graphite and conductive (WC) powder samples. The SPS bulk and contact multiphysics phenomena arising during sintering of ultrafine binderless WC powders are finally discussed. A direct correlation between sintering microstructure, sintering temperature, and heating rate is established. The developed self-consistent SPS model can be effective used as an aiding tool to design optimum SPS experiments, predict sintering microstructure, or benchmark SPS system hardware or performances.     

On the contact force distributions of granular mixtures under 1D-compression

We investigate the distribution of the inter-particle contact forces inside granular mixtures of a sand-particle-size material and of a finer-particle-size material using the Discrete Element Method for frictional spherical grains. The numerical granular samples were compressed vertically with no lateral expansion following a common stress path in soil mechanics; the material states varied from jammed states towards highly jammed states with increasing solid fraction. The inter-particle contacts were categorized depending on the particle sizes of the two contacting entities. The force distributions of the contact networks were calculated depending on the contact types. It was found that different contact networks possess a similar shape of the probability distribution function of the contact forces when the populations of the respective particle sizes are involved in the percolation of the strong forces in the systems. For systems of a small percentage of the fine particles, the fine particles do not actively participate in the strong force transmission and the related contact force distributions reflect the characteristic of an unjammed state for the subsystem consisted of these particles.     

粉末高速压制成形密度分布的数值模拟及影响因素分析

将研究不连续体力学行为的离散单元法应用于粉末高速压制致密化过程的研究,将粉末视为黏弹性的离散颗粒,建立粉末高速压制过程颗粒接触模型及每个颗粒的基本运动方程,推导了力与位移表达的粉末高速压制黏弹性本构关系。基于PFC软件实现了铁粉高速压制过程中粉末颗粒二维流动情况及压坯密度分布的数值模拟,模拟结果的密度分布规律与实际压制的密度分布规律较为一致;利用数值模拟结果对影响压坯密度分布的摩擦因数、高径比、双向压制因素进行了具体分析。     

Ultrasonic Characterization of Iron Powder Metallurgy Compacts during and after Compaction

Ultrasonic measurements in powder metallurgy (PM) compacts at various stages of production are presented both as a practical means of improving PM production and as a method of providing a fuller understanding of PM materials. Ultrasonic monitoring during powder compaction, a novel process instrumentation technique to follow powder densification, is reviewed. Measurements taken during the compaction of simple PM disk demonstrate that the ultrasonic velocity can be used as a measure of the in situ density. This connection arises due to the acoustic equivalence between powder during compaction and PM compacts after sintering. Ultrasonic monitoring during compaction of a two-level PM part is demonstrated to be fully capable of independently following the density in each level. The results also provide evidence of different regimes of powder flow behaviour during compaction. Ultrasonic velocity mapping of the two-level compact after sintering provides confirmation of the monitoring results. Subsequently, measurements of the ultrasonic velocity in green PM compacts are shown to be consistent with a dependence on the quality of inter-particle bonding. Finally, laser ultrasonic measurements in PM compacts are used to determine the ultrasonic attenuation. Attenuation values in a sintered compact are shown to follow a simple Rayleigh scattering dependence on frequency which yields a powder particle size consistent with the known value.     

Modeling forces generated within rigid liquid composite molding tools. Part B: Numerical analysis

Liquid composite molding (LCM) processes generate forces on tooling due to internal resin pressure fields and the resistance to compaction offered by fiber reinforcements. In Part A of this work the authors have presented a detailed study on the evolution of total clamping force during resin transfer molding (RTM) and injection/compression molding (I/CM) cycles. The influence of the complex compaction response of two different reinforcements was demonstrated, important effects including stress relaxation, an apparent lubrication by the injected fluid, and permanent deformation. In the current paper attempts are made to model clamping force evolution utilizing elastic reinforcement compaction models. The predictions are shown to have significant qualitative errors if a single elastic model is applied, particularly if forces due to reinforcement compaction dominate those due to fluid pressure. By using a combination of elastic models significant qualitative and quantitative improvements were made to the predictions. It is concluded that careful characterization of both reinforcement permeability and compaction response are required for an accurate LCM tooling force analysis.