# Advanced Estimation of Compressive Strength and Fracture Behavior in Ceramic Honeycombs by Polarimetry Measurements of Similar Epoxy Resin Honeycombs

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{σ}and the stress can be calculated using Equation (2) [26,28]:

## 2. Materials and Methods

#### 2.1. Sample Preparation

^{®}028J, with the resin Fusia DC700 (both: DWS S.r.l., Zanè, Italy) and a z-layer resolution of 20 µm. These were then molded with polydimethylsiloxane (PDMS) (Elastosil M 4643 A/B, Wacker Chemie AG, München, Germany) to obtain the negative silicon casting molds for the transfer-molding process. The alumina feedstock contained 53 vol% alumina CT3000SG (Almatis GmbH, Ludwigshafen, Germany), 40.4 vol% paraffin wax (Granopent P, Carl Roth GmbH, Karlsruhe, Germany) and 4.6 vol% carnauba wax (Naturfarben, Carl Roth GmbH, Karlsruhe, Germany). Transfer molding was performed at 120 °C supported by applying a moderate vacuum (<10 Pa). Debinding (600 °C) and sintering (1600 °C, 2 h) were performed on porous mullite substrates (Annamullit

^{®}88, Compagnie de Saint-Gobain S.A., Courbevoie, France) with adapted heating and cooling rates between 0.1 and 5 K/min.

#### 2.2. Characterization

## 3. Results and Discussion

#### 3.1. Influence of the θ-Angle on the Photoelastic Stresses

^{−1}was obtained via Equation (1) (glass 2–4 TPa

^{−1}, epoxy resins between 5 and50 TPa

^{−1}) [26,35,36]. Based on the number of fringes and the stress optical coefficient, the internal stress was determined to be 4.9 MPa per fringe (Equation (2)). To count the fringes, it was necessary to know the sources drawn in the stress-optical images in Figure 3 since the fringes originated from these points, and thus represented the highest order and most-elevated stress. A stress band was formed between the source points with a saddle point in the middle, which has the lowest stress. Perpendicular to the band, the internal stresses decreased again. This band was visible at large negative angles (Figure 3) of −35° and −25° between points B and F, which was the preferred fracture direction. [37,38,39] The stress pattern changed significantly with an increasing angle and showed a linear horizontal and vertical stress alignment, especially between −5° to 5°. With a rising θ-angle over 15°, the overall stress rises again and becomes more inhomogeneous.

#### 3.2. Structural Influence on Mechanical Properties

_{0}, which is 2650 MPa for alumina [29]. Here, the exact geometric values of each sample were used to generate a standard deviation of the theoretical calculation. At an angle of 0°, no strength can be calculated, as the function tends to infinity at θ = 0°:

_{MPS}) and the alumina’s compressive strength (σ

_{c}) is shown in Figure 5b and can be described by Equation (4). This is an adaptation of the Gibson and Ashby model (Equation (3)). The median photoelastic stress replaces the angle θ, and the structural parameters are replaced by the empirical constants b and c. The fitted constants in the auxetic range are b = 3.82 MPa and c = −4.5 MPa by R

^{2}= 0.92 and for the hexagonal b = 1.27 MPa and c = −13.92 MPa by R

^{2}= 0.97. In both cases, the model is suitable for predicting the strength of ceramics from photoelastic measurements of epoxy resin. This also demonstrates that the properties are purely influenced by the structure, and they are independent of the materials if they show the same fracture and stress behaviors.

## 4. Conclusions

- Hexagonal and auxetic honeycomb unit cells were fabricated from epoxy and alumina with an θ-angle of −35 to +35;
- Critical stress points were identified by photoelastic measurements of epoxy, which closely matched those fracture points in the alumina determined by DIC;
- Smaller absolute angles showed more homogeneous stress distributions, which were also reflected in the compressive strengths of the alumina, with a maximum of 446 ± 156 MPa at 0°
- The most important achievement was the correlation of the photoelastic measurement of the epoxy with the compressive strength of the alumina by adapting the model from Gibson and Ashby.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) Auxetic and (

**b**) hexagonal unit cell with symmetry axes (X/Y) and structural parameters: strut thickness t, width h, angle θ, leg length l, and measuring points A−F for photoelastic stress measurement, (

**c**) alumina unit cells with varying angle θ between θ = −35° and 35°.

**Figure 2.**Photoelastic characterization of three rectangular epoxy resin references (27 × 3.15 × 3.15 mm

^{3}) to measure the retardation dependent on the applied load. Determination of the gradient via a linear fit to calculate the stress-optical coefficient via Equation (1).

**Figure 3.**The stress-optical rings (numbered in ascending order) of polymer unit cells are shown for angles θ = −35° and 35°. In the magnification of θ = −35°, the starting points B and F of the rings and the saddle point between both rings are shown. The lowest stress occurred at the saddle point.

**Figure 4.**(

**a**) Photoelastic stress at defined measurement points for the epoxy resin unit cells depending on the θ-angle (θ ≥ 0 hexagonal, θ < 0° auxetic). (

**b**) Compressive strength of alumina ceramic unit cells dependent on the θ-angle and the median photoelastic stresses in the epoxy resin unit cells.

**Figure 5.**(

**a**) Main initial fracture in ascending probability of the alumina samples characterized by DIC (

**b**). Compressive strength of ceramic unit cells dependent on the median photoelastic stress of the epoxy unit cells and fitted correlation by an adapted Gibson and Ashby (GA) model: Equations (3) and (4).

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**MDPI and ACS Style**

Köllner, D.; Tolve-Granier, B.; Simon, S.; Kakimoto, K.-i.; Fey, T.
Advanced Estimation of Compressive Strength and Fracture Behavior in Ceramic Honeycombs by Polarimetry Measurements of Similar Epoxy Resin Honeycombs. *Materials* **2022**, *15*, 2361.
https://doi.org/10.3390/ma15072361

**AMA Style**

Köllner D, Tolve-Granier B, Simon S, Kakimoto K-i, Fey T.
Advanced Estimation of Compressive Strength and Fracture Behavior in Ceramic Honeycombs by Polarimetry Measurements of Similar Epoxy Resin Honeycombs. *Materials*. 2022; 15(7):2361.
https://doi.org/10.3390/ma15072361

**Chicago/Turabian Style**

Köllner, David, Bastien Tolve-Granier, Swantje Simon, Ken-ichi Kakimoto, and Tobias Fey.
2022. "Advanced Estimation of Compressive Strength and Fracture Behavior in Ceramic Honeycombs by Polarimetry Measurements of Similar Epoxy Resin Honeycombs" *Materials* 15, no. 7: 2361.
https://doi.org/10.3390/ma15072361