Continuum Theory Of Sintering Research Articles

Complex shape transparent Al2O3 fabricated by integrated Spark Plasma Sintering - Additive manufacturing technology

Spark plasma sintering (SPS) is one of the most promising technologies for producing polycrystalline transparent ceramics. However, it has limitations regarding the complexity of the geometries that can be produced. This study addresses these limitations by combining additive manufacturing techniques with spark plasma sintering. Additionally, a Multiphysics sintering model based on the continuum theory of sintering is employed to predict the outcomes of the SPS process, particularly the microstructure of the densified parts. As a result, the geometrically complex Al2O3 transparent parts manufactured in this study exhibit a uniform microstructure, high density (∼99 %), and a linear transmittance of 16 % at 490 μm.

Mitigation of gravity-induced distortions of binder-jetting components during rotational sintering

Using theory and simulations, the challenge of gravity-induced distortions during sintering is addressed and a mitigation strategy is proposed. Based on the continuum theory of sintering, the finite element simulation demonstrates the advantages of a rotating furnace to counteract gravity forces during sintering. Its application for stainless steel hollow parts produced by additive manufacturing (binder jetting) is demonstrated, numerically, for reliable industrial production of complex shapes. Sintering a tube in a very slow rotating motion exhibits an improvement in the final deformation ratio compared to a conventional sintering process.The same concept has been adapted for higher furnace revolution speeds and the centrifugal force is now surpassing the effects of gravity. An extended study of sintering under microgravity for space-borne applications is also widely depicted with the same model. Indeed, it shows the possibility of reproducing Earth's sintering conditions at places where gravity is insufficient to provide acceptable densification and shape conservation during sintering.

3D Visualization of Morphological Evolution of Large Defects during Spark Plasma Sintering of Alumina Granules

The mechanical reliability of products must be assured for scaling up and production of complex‐shaped components by spark plasma sintering (SPS) of spray‐dried granules. The evolution of morphologies of pores and defects, which control the mechanical strength, is investigated by using synchrotron X‐ray multiscale tomography during SPS of alumina granules at 1300 °C. While large defects arising from the hierarchical granule packing structure cannot be removed by pressureless sintering, crack‐like defects and branched rodlike defects are almost eliminated by SPS at stresses higher than 30 and 50 MPa, respectively. But, small ellipsoidal porous regions, which may arise from aggregates or dimples of granules, cannot be removed even at a pressure of 50 MPa. A very large defect is also found by using micro‐CT. It is supposed that this defect is formed from a large void in loosely packed granules. The shrinkage of large voids and the elimination of crack‐like defects are explained by the theoretical prediction based on the continuum theory of sintering.

Sintering dilatometry based grain growth assessment

Abstract Volume shrinkage, grain growth, and their interaction are major events occurring during free sintering of ceramics. A high temperature sintering dilatometry curve is influenced by these both phenomena. It is shown that the continuum theory of sintering can be utilized in the format enabling the extraction of the maximum amount of information on the densification and grain growth kinetics based on a simple dilatometry test. We present here the capability of such a fast approach (Dilatometry based Grain growth Assessment DGA) utilized for the modeling of sintering and grain growth of zirconia.

Effects of loading modes on densification efficiency of spark plasma sintering: sample study of zirconium carbide consolidation

Theoretical studies on the densification kinetics of the new spark plasma sinter-forging (SPS-forging) consolidation technique and of the regular SPS have been carried out based on the continuum theory of sintering. Both modelling and verifying experimental results indicate that the loading modes play important roles in the densification efficiency of SPS of porous ZrC specimens. Compared to regular SPS, SPS-forging is shown to be able to enhance the densification more significantly during later sintering stages. The derived analytical constitutive equations are utilised to evaluate the high-temperature creep parameters of ZrC under SPS conditions. SPS-forging and regular SPS setups are combined to form a new SPS hybrid loading mode with the purpose of reducing shape irregularity in the SPS-forged specimens. Loading control is imposed to secure the geometry as well as the densification of ZrC specimens during hybrid SPS process.

Fully coupled electromagnetic‐thermal‐mechanical comparative simulation of direct vs hybrid microwave sintering of 3Y‐ZrO2

AbstractDirect and hybrid microwave sintering of 3Y‐ZrO2 are comparatively studied at frequency of 2.45 GHz. Using the continuum theory of sintering, a fully coupled electromagnetic‐thermal‐mechanical (EMTM) finite element simulation is carried out to predict powder samples deformation during their microwave processing. Direct and hybrid heating configurations are computationally tested using advanced heat transfer simulation tools including the surface to surface thermal radiation boundary conditions and a numeric proportional‐integral‐derivative regulation (PID). The developed modeling framework shows a good agreement of the calculation results with the known experimental data on the microwave sintering of 3Y‐ZrO2 in terms of the densification kinetics. It is shown that the direct heating configuration renders highly hot spot effects resulting in nonhomogenous densification causing processed specimen's final shape distortions. Compared with the direct heating, the hybrid heating configuration provides a reduction of the thermal inhomogeneity along with a densification homogenization. As a result of the hybrid heating, the total densification of the specimen is attained without specimen distortions. It is also shown that the reduction of the sample size has a stabilization effect on the temperature and relative density spatial distributions.

Sintering of bi-porous titanium dioxide scaffolds: Experimentation, modeling and simulation

Highly bimodal pore size distributions are found in ceramic scaffolds prepared by freeze casting (ice templating). The scaffolds, with preferred orientation in the ice crystal growth direction, have large interlamellar pores between the scaffold walls and small intergranular pores within the walls. The constitutive equations based on a continuum theory of sintering are used in a finite element analysis of the sintered scaffolds. The calculations employed a model of large isolated pores surrounded by a shell containing small pores. During sintering at 1173 to 1773K the scaffold walls densified first, followed by wall closure at higher temperatures. Excellent agreement is found between the predicted and measured porosity. The compressive strength and elastic modulus depend on the fraction of interlamellar porosity, not the total porosity, indicating that the scaffold wall strength does not affect these properties.

Finite Element Modeling of Camber Evolution During Sintering of Bilayer Structures

The need for understanding the mechanisms and optimization of shape distortions during sintering of bilayers is necessary while producing structures with functionally graded architectures. A finite element model based on the continuum theory of sintering was developed to understand the camber developments during sintering of bilayers composed of La0.85Sr0.15MnO3 and Ce0.9Gd0.1O1.95 tapes. Free shrinkage kinetics of both tapes were used to estimate the parameters necessary for the finite element models. Systematic investigations of the factors affecting the kinetics of distortions such as gravity and friction as well as the initial geometric parameters of the bilayers were made using optical dilatometry experiments and the model. The developed models were able to capture the observed behaviors of the bilayers’ distortions during sintering. Finally, we present the importance of understanding and hence making use of the effect of gravity and friction to minimize the shape distortions during sintering of bilayers.

Kinetics of shrinkage and shape evolution during liquid phase sintering of tungsten heavy alloy

A numerical model describing gravity-induced shape distortion and densification during solid and liquid phase sintering is proposed. The constitutive formulation is based on the continuum theory of sintering and implemented in commercial finite element software. Simulations under gravity-induced stress are attempted on the basis of the model parameters where viscosity is assumed to be temperature and porosity dependent. Viscosity is assessed through shrinkage and shrinkage rate data obtained experimentally from dilatometry over a controlled temperature regime. Effects of temperature, heating rate, and liquid phase formation on porosity evolution are analyzed. Additionally, sample studies on the influence of heating rate, gravity, friction coefficient, aspect ratio, and volume on the predicted distortion profiles after sintering of a tungsten heavy alloy are also presented. These numerical results are compared with experimental data from the literature.

Sintering of Multilayered Porous Structures: Part I‐Constitutive Models

Theoretical analyses of shrinkage and distortion kinetics during sintering of bilayered porous structures are carried out. The developed modeling framework is based on the continuum theory of sintering; it enables the direct assessment of the cofiring process outcomes and of the impact of process controlling parameters. The derived “master sintering curve”‐type solutions are capable of describing and optimizing the generic sintering shrinkage and distortion kinetics for various material systems. The approach utilizes the material‐specific parameters, which define the relative kinetics of layer shrinkages such as the relative intensity of sintering, and employs the conversion between real and specific times of sintering. A novel methodology is also developed for the determination of the ratio of the shear viscosities of the layer's fully dense materials. This new technique enables the determination of all input parameters necessary for modeling sintering of bilayers using experimental techniques similar to optical dilatometry applied to each individual layer and to a symmetric trilayered porous structure based on the two‐layer materials utilized in the bilayered system. Examples of sintering different porous bilayered systems are presented to justify the capability of the model in predicting and optimizing sintering kinetics.

Modeling of gravity-induced shape distortions during sintering of cylindrical specimens

A finite element approach for modeling the gravity-affected sintering process is presented. The constitutive equations are based on the continuum theory of sintering in the framework of a linear viscous material behavior. The model describes the gravity influence on porosity evolution and shrinkage inhomogeneity. Simulations of densification, shape distortion, and porosity gradients are presented. The results are compared with a previously developed analytical model of sintering under the influence of gravity. First time a direct assessment of the impact of the densification inhomogeneity on the gravity-induced shape distortion during sintering is provided in a generic form similar to the master sintering curve approach.

Modeling kinetics of distortion in porous bi-layered structures

Abstract Shape distortions during constrained sintering experiment of bi-layer porous and dense cerium gadolinium oxide (CGO) structures have been modeled. Technologies like solid oxide fuel cells require co-firing thin layers with different green densities, which often exhibit differential shrinkage because of different sintering rates of the materials resulting in undesired distortions of the component. An analytical model based on the continuum theory of sintering has been developed to describe the kinetics of densification and distortion in the sintering processes. A new approach is used to extract the material parameters controlling shape distortion through optimizing the model to experimental data of free shrinkage strains. The significant influence of weight of the sample (gravity) on the kinetics of distortion is taken in to consideration. The modeling predictions indicate good agreement with the results of sintering of a bi-layered CGO system in terms of evolutions of bow, porosities and also layer thickness.

Continuum Isotropic Elastic Model for Sintering of Ceramic Materials <sup></sup>

In this work a sintering model that assumes infinitesimal displacements rates has been structured using the continuum theory of sintering develop by Olevsky and Skorokhod [1, 2] and the sintering model develop by Sasan Kiani et al. [3, 4]. The model was used to estimate dimensional change and the density distribution using linear displacement rates (LRDs) as the only input. Furthermore, if the sintering potential and dense material shear viscosity are known, the stress distribution can be approximated. The magnitude of two stress norms (i.e. Von Misses stress and the stress tensor first invariant) determined using experimental and calculated LDRs are compared using three different geometries. From FEM simulation results it was inferred that densification can occur in presence of tension stresses if their magnitude is lower than the calculated sintering potential.

Sintering of hybrid skeletal materials containing voids filled with powder

The sintering behavior of a skeletal material whose cells contain powder is analyzed. The analysis is based on the simulation of this process using the continuum theory of sintering. The calculations are performed with the finite-element method. Special attention is paid to the distortion of cells and evolution of porosity distribution within the cells during sintering. If the powder and skeleton are bonded, combinations of cell and wall sizes and initial porosity are such that internal cavities can form between the powder and skeleton walls after sintering.

Graded Powder Composites by Freeze Drying, Electrophoretic Deposition and Sintering

Two approaches for the fabrication of tailored powder composites with specially distributed pore-grain structure and chemical composition are investigated. Electrophoretic Deposition (EPD) followed by microwave sintering is employed to obtain functionally graded materials (FGM) by in-situ controlling the deposition bath suspension composition. Al2O3/ZrO2 and zeolite FGM are successfully synthesized using this technique. In order to fabricate an aligned porous structure, unidirectional freezing followed by freeze drying and sintering is employed. By controlling the temperature gradient during freezing of powder slurry, a unidirectional ice-ceramic structure is obtained. The frozen specimen is then subjected to freeze drying to sublimate the ice. The obtained capillary-porous ceramic specimen is consolidated by sintering. The sintering of the graded structure is modeled by the continuum theory of sintering.

Kinetics and stability in compressive and tensile loading of porous bodies

The continuum theory of sintering is used for the analysis of the kinetics of compressive and tensile loading (forging, drawing, isostatic pressing, etc.) of a cylindrical powder specimen. The constitutive properties of the powder material are assumed to be nonlinear viscous, corresponding to the power-law creep mechanism. The calculations are conducted for various models developed for the description of the deformation processing of viscous porous materials. The calculation results for uniaxial compression are compared with the experimental data on forging of stainless steel 304 porous specimens. Both linear and nonlinear stability analyses are carried out. Stability criteria are derived for uniform and nonuniform porosity perturbation modes and for two different kinds of boundary conditions: constant axial force and constant imposed cross-head velocity under uniaxial loading. Stability maps are obtained for forging of copper powder components.

Stability analysis for forging of porous bodies

The continuum theory of sintering is used for the analysis of the stability of forging of a cylindrical powder specimen. The constitutive properties of the powder material are assumed to follow a power-law creep relationship. Temperature-coupled linear non-uniform stability analyses are carried out. Stability maps are obtained for forging of copper powder components.

Effect of gravity on dimensional change during sintering—I. Shrinkage anisotropy

The effects of gravity on sintering shrinkage and dimensional uniformity are analyzed using a continuum theory of sintering. Shape change caused by gravity during sintering is described both analytically and numerically. For a cylindrical sample shape, analytical approximations to characterize gravity-induced nonuniformities in shrinkage and compact aspect ratio are shown. As an illustration of the concept, analysis is performed for the solid-phase sintering of a copper powder cylindrical specimen to replicate earlier experiments reported by Lenel et al. ( Trans. Am. Inst. Min. Engrs, 1963, 227, 640) and Exner ( Rev. Powder. Metall. Phys. Ceram. Soc., 1968, 51, 604). The intensity of shrinkage anisotropy is compared for viscous and diffusional mechanisms of sintering. An algorithm is introduced for minimization of gravity-induced shrinkage anisotropy, suggesting an asymptotic approach to the peak temperature for best dimensional uniformity.

Effect of gravity on dimensional change during sintering—II. Shape distortion

Phenomena of shape distortion caused by gravity under solid- and liquid-phase sintering are described numerically using the finite-element method based upon a continuum theory of sintering. Finite-element analysis is performed for the solid-phase sintering of a copper powder cylindrical specimen to replicate earlier experiments reported by Lenel et al ( Trans. Am. Inst. Min. Engrs, 1963, 227, 640). The results of the finite-element calculations are also compared with experimental data on liquid-phase sintering of a W–Ni–Fe powder system using a quantitative criterion of shape distortion. It is demonstrated that shape distortion accumulated during solid-phase sintering should be taken into account when considering gravity-induced configuration—dimensional changes under liquid-phase sintering. The finite-element analysis of shape distortion under liquid-phase sintering reproduces an “elephant foot” shape—a final shape observed in sintering practice.

Theory of sintering: from discrete to continuum

Theoretical concepts of sintering were originally based upon ideas of the discrete nature of particulate media. However, the actual sintering kinetics of particulate bodies are determined not only by the properties of the particles themselves and the nature of their local interaction with each other, but also by macroscopic factors. Among them are externally applied forces, kinematic constraints (e.g. adhesion of the sample's end face and furnace surface), and inhomogeneity of properties in the volume under investigation (e.g. inhomogeneity of initial density distribution created during preliminary forming operations). Insufficient treatment of the questions enumerated above was one of the basic reasons hindering the use of sintering theory. A promising approach is connected with the use of continuum mechanics, which has been successfully applied to the analysis of compaction of porous bodies. This approach is based upon the theories of plastic and nonlinear-viscous deformation of porous bodies. Similar ideas have recently been embodied in a continuum theory of sintering. The main results of the application of this theory for the solution of certain technological problems of sintering are introduced including their thermo–mechanical aspects.