# Mesoscale Models for Describing the Formation of Anisotropic Porosity and Strain-Stress Distributions during the Pressing Step in Electroceramics

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## Abstract

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## 1. Introduction

_{3}ceramics with variable relative porosity levels (in the range of 8% to 26%) produced by using Poly(methyl methacrylate) (PMMA) spherical inclusions as sacrificial phase, have been analyzed. These ceramics were obtained by sintering of uniaxially pressed mixtures of oxide and polymeric powders. Even if it is clear that the final microstructures resulted after sintering are different from those of the green ceramics obtained after the pressing step (due to the processes of shrinkage, densification, grain growth, small porosity elimination, etc. taken place during the sintering), the elongated shape of the observed pores suggested that a plastic deformation of the soft equiaxed polymeric particles took place during the pressing.

## 2. Materials and Methods

_{3}oxide powders monodisperse Poly(methyl methacrylate) (PMMA) microspheres with diameters of ~10 μm (Figure 1a). The mixed powders have been coldly pressed at 150 MPa (uniaxially or isostatic) and then, the green ceramics have been densified by applying a pressureless multi-step thermal treatment to ensure the combustion of polymeric particles, the complete gas elimination before closing the ceramic pores and complete sintering, as described in detail in ref. [6]. The surface ceramic morphology has been examined in fresh ceramic fractures by high resolution scanning electronic microscopy (SEM) with a Carl Zeiss System NEON40EsB. In order to better observe the 3D pore morphology in the sample, 3D micro X-ray computed tomography (μ-XCT) [22,23,24] with a nanotom m from General Electrics (GE) was pursued. The voxel size was set to 1.33 µm and the selected volume of interest (VOI) for the analysis was 186 × 186 × 186 μm

^{3}big. Python libraries (including Numpy 1.18.1, Scipy 1.4.1, Scikit-image 0.16.2, Matplotlib 3.1.3, and Scikit-fmm 2019.1.30) and Avizo

^{®}ThermoFischer Scientific, version 2019.1 and 2021.2 have been used to perform the image processing and visualization of the µ-XCT data. In order to compute the strain-stress fields and deformation of such structures, containing a maximum amount of 950 equiaxed soft inclusions in a hard matrix, some numerical methods based on the commercial finite element codes (COMSOL Multiphysics v. 6.0, COMSOL Inc., Burlington, VT, USA) have been used. The statistical analysis to determine the angular distribution and deformability level, as result of the isostatic and uniaxial pressing was performed by using C/C++ programs.

## 3. Results and Discussion

#### 3.1. Microstructural Characterization

_{3}ceramics with a relative porosity of about 19%, uniaxially pressed (Figure 2a) and with a relative porosity of about 13%, isostatically pressed (Figure 2b) indicate the presence of certain anisotropy resulted by both types of pressing, when using spherical sacrificial polymeric additions.

#### 3.2. Modelling

#### 3.2.1. Analytical Approach

^{−8}mm. Other important limitations are determined from the mathematical apparatus itself considering for calculations a single soft inclusion and from the effects of the top and bottom flat surfaces on the nearby material, which cannot be determined by using only the Hooke’s law. Another drawback is determined by the impossibility of studying the interaction between more soft inclusions and their geometrical boundaries. Overall, those results are promising because the deformed shape of the outermost inclusion matches the observed shape of a squeezed hollow sphere near the boundary of a solid cylinder, but these overall observations suggest that a numerical model would be more appropriate.

#### 3.2.2. Numerical Approach by Finite Element Modelling

**A.****Isostatic pressing**

**B.****Uniaxial pressing**

## 4. Conclusions

_{3}ceramics when using as pore forming agent polymeric spherical particles is investigated. The presented model works equally well for any matrix material, as long as it can be considered elastic, homogeneous and isotropic. Using analytical and numerical calculations it is shown that during the pressing step of a mixture of solid particles characterised by contrasting mechanical properties (hard/soft), the shape of the soft deformable phase, initially equiaxed changes both when using uniaxial or isostatic pressing. In both cases, anisotropic distribution of such inclusions in the green body resulted. A higher shape homogeneity and a broad bi-modal angular distribution of the elongated inclusions along two perpendicular main axes resulted by simulations for isostatic pressing. In contrast, a broad log-normal distribution of the inclusion’s aspect ratio showing a stronger shape inhomogeneity and a high degree of anisotropy resulted for the uniaxial pressing. Starting with such green ceramic microstructures, the deformed inclusions would generate after sintering elongated and anisotropic porosity microstructures and anisotropic related functional properties. This type of porosity originated from the soft polymeric pyrolisable additives is different than the naturally occurred one in a sintered ceramic, as resulted by incomplete sintering (which is isotropic and pores are equiaxed). The first part of this study using an analytical approach for a single equiaxed inclusion into a continuum matrix was able to give information concerning the tendency of the soft phase to elongate along the symmetry axis, thus, resulting in a strong anisotropy in one direction. The finite element numerical approach performed in 2D planes along transversal and radial directions indicated the anisotropy of the soft phase along two main directions, symmetrical with respect to the major cylinder axis (i.e., bimodal distributions). For the two types of pressing, elongated soft inclusions are found, with different shape factor distributions and anisotropy. The numerical approach can be further improved by considering non-linear mechanical characteristics for the ceramic powders, coupling the elastic properties with the Drucker-Prager theory in order to better understand the consolidation step and by building a 3D model able to consider more elastic effects taking place in the bulk matrix.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**SEM micrographs in fractured fresh surfaces of the PMMA spheres used as pore forming agent (

**a**) and for the sintered BaTiO

_{3}ceramics with elongated porosity (~10 vol.%) obtained by the addition of PMMA spheres uniaxially pressed (

**b**), or isostatically pressed (

**c**) and for a BaTiO

_{3}ceramic realized without any polymeric addition, showing natural occurred spherical porosity (4 vol.%), as result of imperfect sintering (

**d**).

**Figure 2.**3D images of the processed BaTiO

_{3}ceramics structures derived by micro X-ray computed tomography of with a relative porosity of ~19%, uniaxially pressed (

**a**) and with a relative porosity of ~13%, isostatically pressed (

**b**); (

**c**,

**d**) Correlation functions along the main local axes for the structures shown in the figures (

**a**,

**b**).

**Figure 3.**Illustration of the main stresses in a cylindrical body (

**a**); radial and tangential stresses on an insulated volume element (

**b**); calculation of the external radius: $\overrightarrow{{R}_{e}}=\overrightarrow{OA}-\overrightarrow{OB}$ (

**c**); illustration of the mechanical equilibrium conditions (

**d**).

**Figure 4.**Position of the presented inclusions relative to the blue cylinder axis Oz: $r$ = 0 (

**a**), 1.47 mm (

**b**), 2.93 mm (

**c**) and 4.4 mm (

**d**).

**Figure 5.**Systems with 10, 50, 500 and 950 circular inclusions (

**a**) and the corresponding simulated microstructures after the deformation by isostatic pressing (

**b**).

**Figure 6.**Schematic of isostatic pressing: (

**a**) pressure applied only along the cylinder axis, (

**b**) after isostatic pressing and deformation; (

**c**) isostatically deformed structure; (

**d**) angular distribution of inclusions derived for the case of isostatic pressing.

**Figure 7.**Initial (

**a**) and deformed (

**b**) shape of the soft inclusions in the transversal section for the case of isostatic pressing; (

**c**) detail of the deformed structure and von Mises stresses; (

**d**) statistical angular distribution in the transversal section.

**Figure 8.**Schematic of uniaxial pressing: (

**a**) before pressing, (

**b**) after pressing and deformation; (

**c**) uniaxially entirely deformed structure; (

**d**) angular distribution of inclusions derived for uniaxial pressing.

**Figure 9.**Comparative plot of the aspect ratio distributions of the soft phase for the isostatic and uniaxial pressing.

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**MDPI and ACS Style**

Stirbu, R.S.; Padurariu, L.; Chamasemani, F.F.; Brunner, R.; Mitoseriu, L.
Mesoscale Models for Describing the Formation of Anisotropic Porosity and Strain-Stress Distributions during the Pressing Step in Electroceramics. *Materials* **2022**, *15*, 6839.
https://doi.org/10.3390/ma15196839

**AMA Style**

Stirbu RS, Padurariu L, Chamasemani FF, Brunner R, Mitoseriu L.
Mesoscale Models for Describing the Formation of Anisotropic Porosity and Strain-Stress Distributions during the Pressing Step in Electroceramics. *Materials*. 2022; 15(19):6839.
https://doi.org/10.3390/ma15196839

**Chicago/Turabian Style**

Stirbu, Radu Stefan, Leontin Padurariu, Fereshteh Falah Chamasemani, Roland Brunner, and Liliana Mitoseriu.
2022. "Mesoscale Models for Describing the Formation of Anisotropic Porosity and Strain-Stress Distributions during the Pressing Step in Electroceramics" *Materials* 15, no. 19: 6839.
https://doi.org/10.3390/ma15196839