Constitutive modeling of the behaviour of cermet compacts during reaction sintering

This study deals with the identification of a constitutive equation describing the mechanical behaviour of a nickel ferrite based cermet during sintering. This constitutive equation considers the material as a continuum and may enable one to predict the densification behaviour of a powder under different thermal treatments and the impact of compact geometry, external loading on strain and stress generation. A classical viscous equation of the Newtonian type that includes a term describing free sintering densification has been chosen. The method used for the identification of the parameters of this equation is the one proposed Gillia et al., which is based on dilatometry measurement. It includes a stairway thermal cycle for the determination of the free sintering term and intermittent loading for estimating the viscosity. This approach has been successfully applied to nickel ferrite cermet. The model has been found to be adequate to model the densification behaviour up to 1250 °C, but experimental and theoretical efforts must be accomplished to describe the behaviour above this temperature, when the material exhibits swelling.     

Evaluation of dynamic viscosity and rheological properties of ceramic materials during liquid phase sintering by numerical‐experimental procedure

The effective shear and bulk viscosity, as well as dynamic viscosity, describe the rheological properties of the ceramic body during the liquid phase sintering process. The rheological parameters depend on the physical and thermo‐mechanical characteristics of the material such as relative density, temperature, grain size, diffusion coefficient, and activation energy. In this paper, the numerical‐experimental method has been developed to study both viscous and rheological behavior of hard porcelain ceramic body during liquid phase sintering. The other aim is to acquire a complete understanding of the response of an incompressible viscose material during sintering such as stress‐strain relations, sintering, and hydrostatic stress. Densification results confirmed that the bulk viscosity was well‐defined with relative density. The stress analysis proved that the sintering stress is more than the hydrostatic stress during the entire sintering time so, the sintering process occurs completely. Deflection results showed that the shear viscosity was a fair estimation of real ones. Dilatometry, SEM, XRD investigations as well as bulk viscosity simulation results confirmed that the “mullitisation plateau” was presented as a very little extraordinary expansion at the final sintering stage.     

Drucker-Prager-Cap modelling of boron carbide powder for coupled electrical-thermal-mechanical finite element simulation of spark plasma sintering

The coupled electrical-thermal-mechanical finite element method in the continuum scale has been widely used to investigate the spark plasma sintering process. An accurate constitutive model of powder material is pivotal for precise continuum finite element simulation. In this study, the Drucker-Prager-Cap model, which is highly accurate in describing the densification behaviour of powder material, was adopted to numerically analyse the spark plasma sintering process of boron carbide powder. First, the parameters of the model were defined to be dependent on temperature and density for higher accuracy; they were determined by minimising the discrepancy between the simulated and experimental results. Based on a spark plasma sintering experiment with a cylindrical sample, the parameters of the Drucker-Prager-Cap model were identified at 1500 °C, 1600 °C, 1700 °C, 1800 °C, and 1900 °C. A coupled electrical-thermal-mechanical finite element simulation with the model was performed for spark plasma sintering of boron carbide powder at 1750 °C and 1850 °C. The temperature, stress, and relative density were investigated numerically. By comparing the model results with the temperature and relative density measured in the experiment, the continuum finite element method with the Drucker-Prager-Cap model was validated.     

Modelling of elimination of strength-limiting defects by pressure-assisted sintering at low stress levels

The structural reliability of sintered products depends on large defects introduced during powder processing, which cannot be removed by pressureless sintering. Here, we present a model how a large single ellipsoidal void is deformed, and finally disappears by pressure-assisted sintering. Taya-Seidel’s model is applied to predict the shrinkage of a large void in a compressible linear viscous material by using bulk viscosity, shear viscosity, and sintering stress that are determined experimentally for sintering of alumina powder at low stress levels. The application of mechanical stress promotes the densification rate. Its effect is maximum for hot isostatic pressing (HIP) and minimum for sinter forging. The effect is intermediate for hot pressing (HP) and spark plasma sintering (SPS), because the hydrostatic component of stress varies with densification. While a crack-like defect can be removed during densification, a spherical void must be eliminated by shear deformation in the final stage during dwell time.     

Effect of Green-State Processing on the Sintering Stress and Viscosity of Alumina Compacts

Uniaxial viscosity and sintering stress of pressure filtrated alumina compacts were evaluated from sinter-forging measurements. At a particular density, significantly higher values of sintering stresses are observed compared with specimens made by uniaxial dry pressing followed by cold isostatic pressing. In addition, the uniaxial viscosity at a given density is lower for the pressure-filtrated samples. These differences may be explained by a more homogeneous microstructure and finer pore size and emphasize the importance of green density and packing on the evolution of the constitutive parameters for crystalline materials.     

The Effect of Applied Stress on the Densification of a Low-Temperature Cofired Ceramic-Filled Glass System Under Constrained Sintering

The effect of uniaxial stress on the mechanical response and densification behavior of a low-fire borosilicate glass (BSG)+alumina system during constrained sintering of a multilayer BSG+alumina/alumina laminate has been investigated. Compared with free sintering, the pressure-less constrained sintering of BSG+alumina exhibits poorer densification, and larger porous bulk viscosity at a given temperature. This is caused by the in-plane tensile stress and anisotropic development generated in the transverse directions of the laminate during constrained sintering. The applied uniaxial stress required in the thickness direction to densify BSG+alumina under constrained sintering varies in the range of 50–400 kPa at 700°–800°C. The above results are in agreement with those calculated using the viscous analogy for the constitutive relationships of a porous sintering compact.     

Effect of Rigid Inclusions on the Densification and Constitutive Parameters of Liquid-Phase-Sintered YBa2Cu3O6+x Powder Compacts

The presence of rigid inclusions in a powder compact leads to a reduction in the densification rate of the compact and may also lead to processing defects. In this paper, the densification rate and the constitutive parameters of both homogeneous YBa₂Cu₃O_{6+ x} and composite powder compacts (YBa₂Cu₃O_{6+ x} powder with 10 vol% dense inclusions of YBa₂Cu₃O_{6+ x} ) are reported. A small amount of liquid phase, which formed during sintering, was present in the samples. However, even with the presence of a liquid phase, the addition of inclusions still reduces the densification rate of the composite and increases its viscosity. The results have been compared with a published analysis of the problem using measured values of the constitutive parameters. Both the viscosity and viscous Poisson's ratio of the porous body have been measured.     

Study of cold powder compaction by using the discrete element method

The discrete element method (DEM), based on a soft-sphere approach, is commonly used to simulate powder compaction. With these simulations a new macroscopic constitutive relation can be formulated. It is able to de-scribe accurately the constitutive material of powders during the cold compaction process. However, the force-law used in the classical DEM formulation does not reproduce correctly the stress evolution during the high density compaction of powder. To overcome this limitation at a relative density of about 0.85, the high density model is used. This contact model can reproduce incompressibility effects in granular media by implementing the local solid fraction into the DEM software, using Voronoi cells. The first DEM simulations using the open-source YADE software show a fairly good agreement with the multi-particle finite element simulations and experimental results.     

Constrained Film Sintering of Nanocrystalline TiO2

The sintering of thin, nanocrystalline TiO₂ films either 140 or 65 nm thick is characterized and compared with the sintering of bulk material. Grain size, pore size distribution, and density data are obtained. Observation of the microstructural evolution during sintering suggests that grain growth, as well as pore growth, at low density can be attributed to differential sintering. A continuum mechanical model for the intermediate stage is useful until the grain size becomes one-half the film thickness. From then on, the ratio of grain size to film thickness affects the degree by which both densification and grain growth decrease, as compared with the continuum computation results.     

Cracking and shape deformation of cylindrical cavities during constrained sintering

During constrained sintering of thin films, in which a cylindrical cavity with axis perpendicular to the substrate has been introduced before sintering, cracks emerge that initiate at the cavity surface. By combining experiments with continuum mechanical and particle based simulations, the fundamental causes and effects of this kind of crack formation are identified. A stress analysis performed by finite element (FEM) simulations matches with the cracking behavior observed in experiments. A comparison of discrete element (DEM) results with experiments shows the applicability of this simulation method to describe the effect of cross-sectional stripe dimensions and cavity diameters on the cracking behavior. Moreover, DEM simulations reveal that hair-line cracks in narrow stripe samples formed during pre-sintering manufacturing steps might be a dominant cause for the observed crack damage in such systems.     

Simulation of thermal and electric field evolution during spark plasma sintering

Finite element simulations have been conducted to determine the temperature distribution (both in radial and axial direction), heat and electric flux-field in the powder compact/die/punch assembly during the spark plasma sintering (SPS) process. A thermal–electrical coupled model with temperature dependent thermal and electrical properties is implemented. The simulation studies were conducted using both ABAQUS and MATLAB and a range of power input, varying thermal conductivity of powder compact were considered. Also, the effect of time variation on both thermal and electric field evolution was assessed. During SPS, the heat transmission pattern and the temperature difference between the specimen center and the die surface depend on thermal conductivity of the powder compact, time of sintering and power input. The simulation results also revealed that the temperature gradient across the powder compact/graphite die wall is higher for conditions of higher power input and/or powder compact with lower thermal conductivity.     

Sintering of Spherical Glass Powder under a Uniaxial Stress

The sintering of spherical borosilicate glass powder (particle size 5 to 10 μm) under a uniaxial stress was studied at 800°C. The experiments allowed the measurement of the kinetics of densification and creep, the viscosities for creep and bulk deformation, and the sintering stress which was found to increase with density. The data show excellent qualitative agreement with Scherer's theory of viscous sintering. In addition, the quantitative comparison between theory and experiment shows good agreement; the measured viscosity of the bulk glass was ∽1×10⁹ P (∽1×10⁸ Pa·s) compared to ∽3×10⁹ P (∽3 Pa·s) obtained by fitting the data with Scherer's theory.     

Computation of sintering stress and bulk viscosity from microtomographic images in viscous sintering of glass particles

Sintering stress and bulk viscosity were derived as functions of relative density from microtomographic images in viscous sintering of glass particles. Three methods were proposed to estimate the sintering stress from relative density, specific surface area, and average of curvature on pore surface, which were directly measured by X‐ray microtomography. The surface energy method gave valid value in the final stage of sintering, while the mixed method gave better estimation in the intermediate stage. For the initial stage of sintering, the sintering stress was calculated from the average contact radius and the average coordination number observed by X‐ray microtomography. The sintering stress at the final stage increased in free sintering, but it decreased in constrained sintering due to pore coarsening. The bulk viscosity was calculated from the shrinkage rate and the sintering stress.     

Crack initiation,propagation, and arrest in sintering powder aggregates

Cracking during sintering is a common problem in powder processing and is usually caused by constraint that prevents the sintering material from shrinking in one or more directions. Different factors influence sintering-induced cracking, including temperature schedule, packing density, and specimen geometry. Here we use the discrete element method to directly observe the stress distribution and sinter-cracking behavior in edge notched panels sintered under a uniaxial restraint. This geometry allows an easy comparison with traditional fracture mechanics parameters, facilitating analysis of sinter-cracking behavior. We find that cracking caused by self-stress during sintering resembles the growth of creep cracks in fully dense materials. By deriving the constrained densification rate from the appropriate constitutive equations, we discover that linear shrinkage transverse to the loading axis is accelerated by a contribution from the effective Poisson's ratio of a sintering solid. Simulation of different notch geometries and initial relative densities reveals conditions that favor densification and minimize crack growth, alluding to design methods for avoiding cracking in actual sintering processes. We combine the far-field stress and crack length to compute the net section stress, finding that it characterizes the stress profile between the notches and correlates with the sinter-crack growth rate, demonstrating its potential to quantitatively describe sinter-cracking.     

Simulation of densification behavior of nano-powder in final sintering stage: Effect of pore-size distribution

In the final sintering stage, nano-sized powder frequently forms a pore structure where most pores are surrounded by more than 5 grains. The pore structure is different from that of coarse powder. In this study, the densification behavior of nano-sized powder is modelled and simulated in the final sintering stage. The porous body has the initial size distribution of pores, represented as a Weibull function. The mechanical interaction between pores is analyzed to simulate the evolution of porosity characteristics as well as densification kinetics. The densification rate for the size-distributed pores is lower than that for single-sized ones. The experimental relationship between the densification rate and the porosity could well be reproduced by choosing appropriate pore-size distributions. The simulation also shows that the sintering stress with densification may increase or decrease depending on the size distribution, but is remarkably lower than that for single-sized pores.     

High-pressure compaction modelling of calcite (CaCO3) powder compact

Numerical simulation of manufacturing processes with working conditions at high pressure (above 1 GPa) requires constitutive data of the powder for the whole range of pressure and density. Most of the test apparatuses commonly used to obtain such data are only working in the lower pressure regions. Because of the absence of high-pressure data, many parameters have to be guessed or extrapolated. A material used in high-pressure applications is Calcite (CaCO₃). The material can be used as an insulator in high-pressure capsules it is also a common material in the earth core. An apparatus often used to generate high pressure during compaction is the Bridgman anvil apparatus. In this work experimental tests with a Bridgman anvil set-up using Calcite powder discs with different thicknesses were done. A nonlinear elastic-plastic cap model was developed to model the behaviour of powder material from low pressure and loose state to high pressure and solid state. The constitutive model was implemented in a finite element code. The constitutive data were identified by optimization of experimental data. Validation was done by numerically reproduce the mechanical behaviour of uni-axially pressing Calcite to different pressure (up to 5 GPa) including unloading. The load-displacement curves, density distribution and the surface displacement were measured and compared to the finite element results. The results of the compaction simulations agree reasonably well with the experimental results.     

Tantalum carbide coating via wet powder process: From slurry design to practical process tests

TaC coatings were deposited on graphite substrates via wet powder forming and sintering to potentially achieve both a highly reliable and low-cost process. Non-aqueous solvent mixtures of TaC slurries were optimized through characterization of raw materials and a novel design guide based on Hansen solubility parameters. The optimized TaC slurries enabled the formation of high-quality TaC powder compact films with ultrahigh powder packing densities of ≥70% (ca. 85% at maximum), which contributed to prevent defect formation in the TaC coatings after sintering. The TaC-coated components were tested in practical high-temperature processes, such as AlN and SiC bulk single crystal growth processes, and a SiC device fabrication process, which confirmed sufficiently low-levels of impurity incorporation and surface contamination from the TaC layers. The novel TaC-coated graphite components will contribute to higher quality and lower cost for bulk crystal growth, device fabrication, and epitaxial film growth processes of group-III nitrides and SiC.