# Experimental Investigation of the Tension and Compression Creep Behavior of Alumina-Spinel Refractories at High Temperatures

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## Abstract

**:**

## 1. Introduction

## 2. Material

_{2}O

_{3}, 5% MgO, 0.3% SiO

_{2}and 0.1% Fe

_{2}O

_{3}, with a bulk density of 3.13 g/cm and apparent porosity of 19 vol%.

## 3. Experiments

#### 3.1. Uniaxial Compression Tests

#### 3.2. Uniaxial Tensile Tests

## 4. Methodology

#### 4.1. Inverse Identification

**Step 1**: Definition of the input variables.- Sample’s diameter
- Young’s modulus
- Type of creep (primary or secondary)
- Allowed range of variation for the material’s properties
- Raw data from the tests (time-force-displacement tables)

**Step 2**: Random definition of the initial guesses, depending on the variable’s range of variation and the number of initial guesses.**Step 3**: For each of the initial guesses and each of the stress levels, calculate the analytic time-strain curves using Equation (4), at the same time points as the ones available from the experimental data.**Step 4**: For each time point, calculate the difference between the experimental and analytic values (identification error).**Step 5**: Using a Levenberg-Marquardt optimization algorithm [13], change the material’s parameters in order to minimize the identification error.

#### 4.2. Statistical Analysis

- Statistical population: group of all possible items in the study domain. In the present case, the population is the infinite number of creep tests that could be done.
- Statistical sample: the actual subset of the population being studied. In this study, the statistical sample is used to draw conclusions about the statistical population, since the mean and the standard deviations of the population are unknown.

## 5. Results and Discussions

#### Identification of the Creep Parameters

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Banerjee, S. Properties of Refractories. In Refractories Handbook; Schacht, C.A., Ed.; Marcel Dekker, Inc.: New York, NY, USA, 2004; Chapter 1; pp. 1–10. [Google Scholar]
- Naumenko, K.; Altenbach, H. Modeling of Creep for Structural Analysis; Springer: Berlin/Heidelberg, Germany, 2007. [Google Scholar]
- Jin, S.; Harmuth, H.; Gruber, D. Compressive creep testing of refractories at elevated loads—Device, material law and evaluation techniques. J. Eur. Ceram. Soc.
**2014**, 34, 4037–4042. [Google Scholar] [CrossRef] - Blond, E.; Schmitt, N.; Hild, F.; Blumenfeld, P.; Poirier, J. Modelling of high temperature asymmetric creep behavior of ceramics. J. Eur. Ceram. Soc.
**2005**, 25, 1819–1827. [Google Scholar] [CrossRef][Green Version] - Hynes, A.; Doremus, R. Theories of Creep in Ceramics. Crit. Rev. Solid State Mater. Sci.
**1996**, 21, 129–187. [Google Scholar] [CrossRef] - Martinez, A.T.; Luz, A.; Braulio, M.; Pandolfelli, V. Creep behavior modeling of silica fume containing Al
_{2}O_{3}–MgO refractory castables. Ceram. Int.**2012**, 38, 327–332. [Google Scholar] [CrossRef] - Cannon, W.R.; Langdon, T.G. Creep of ceramics. J. Mater. Sci.
**1983**, 18, 1–50. [Google Scholar] [CrossRef] - de Souza Neto, E.A.; Peric, D.; Owen, D.R.J. Computational Methods for Plasticity; Wiley: West Sussex, UK, 2008. [Google Scholar]
- Mammar, A.S.; Gruber, D.; Harmuth, H.; Jin, S. Tensile creep measurements of ordinary ceramic refractories at service related loads including setup, creep law, testing and evaluation procedures. Ceram. Int.
**2016**, 42, 6791–6799. [Google Scholar] [CrossRef] - Lemaitre, J.; Chaboche, J.L. Mechanics of Solid Materials; Cambridge University Press: Cambridge, UK, 1990. [Google Scholar]
- Schachner, S.; Jin, S.; Gruber, D.; Harmuth, H. Three stage creep behavior of MgO containing ordinary refractories in tension and compression. Ceram. Int.
**2019**, 45, 9483–9490. [Google Scholar] [CrossRef] - Samadi, S.; Jin, S.; Gruber, D.; Harmuth, H.; Schachner, S. Statistical study of compressive creep parameters of an alumina spinel refractory. Ceram. Int.
**2020**. [Google Scholar] [CrossRef] - Marquardt, D.W. An Algorithm for Least-Squares Estimation of Nonlinear Parameters. J. Soc. Ind. Appl. Math.
**1963**, 11, 431–441. [Google Scholar] [CrossRef] - Box, G.E.P.; Hunter, J.S.; Hunter, W.G. Statistics for Experimenters: Design, Innovation, and Discovery; Wiley: Hoboken, NJ, USA, 2005. [Google Scholar]

Parameter | Average | Std. Deviation | 70% Confidence Interval |
---|---|---|---|

${log}_{10}$(A[MPa${}^{-n}$s${}^{-1}$]) | $-13.52$ | $0.925$ | $(-14.08,-12.95)$ |

n [-] | $3.56$ | $0.554$ | $(3.22,3.90)$ |

a [-] | $-2.59$ | $0.218$ | $(-2.73,-2.46)$ |

Parameter | Average | Std. Deviation | 70% Confidence Interval |
---|---|---|---|

${log}_{10}$(A[MPa${}^{-n}$s${}^{-1}$]) | $2.52$ | $1.14$ | $(1.62,3.42)$ |

n [-] | $12.05$ | $1.78$ | $(10.65,13.45)$ |

Parameter | Average | Std. Deviation | 70% Confidence Interval |
---|---|---|---|

${log}_{10}$(A[MPa${}^{-n}$s${}^{-1}$]) | $-14.16$ | $0.506$ | $(-14.49,-13.83)$ |

n [-] | $3.96$ | $0.257$ | $(3.79,4.13)$ |

a [-] | $-2.74$ | $0.142$ | $(-2.83,-2.64)$ |

Parameter | Average | Std. Deviation | 70% Confidence Interval |
---|---|---|---|

${log}_{10}$(A[MPa${}^{-n}$s${}^{-1}$]) | $2.48$ | $0.107$ | $(2.39,2.56)$ |

n [-] | $11.86$ | $0.168$ | $(11.73,11.99)$ |

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## Share and Cite

**MDPI and ACS Style**

Teixeira, L.; Samadi, S.; Gillibert, J.; Jin, S.; Sayet, T.; Gruber, D.; Blond, E. Experimental Investigation of the Tension and Compression Creep Behavior of Alumina-Spinel Refractories at High Temperatures. *Ceramics* **2020**, *3*, 372-383.
https://doi.org/10.3390/ceramics3030033

**AMA Style**

Teixeira L, Samadi S, Gillibert J, Jin S, Sayet T, Gruber D, Blond E. Experimental Investigation of the Tension and Compression Creep Behavior of Alumina-Spinel Refractories at High Temperatures. *Ceramics*. 2020; 3(3):372-383.
https://doi.org/10.3390/ceramics3030033

**Chicago/Turabian Style**

Teixeira, Lucas, Soheil Samadi, Jean Gillibert, Shengli Jin, Thomas Sayet, Dietmar Gruber, and Eric Blond. 2020. "Experimental Investigation of the Tension and Compression Creep Behavior of Alumina-Spinel Refractories at High Temperatures" *Ceramics* 3, no. 3: 372-383.
https://doi.org/10.3390/ceramics3030033