## 3 Results and Discussion
### Strength Analysis  
In Figure 7, there is an SEM image of a sintered sample fracture surface presented. It can be seen that after sintering, the grain size of around 1 \(\upmu\)m was partially retained. Figure 6: The histogram of phase distribution for ZrO\({}_{2}\) stabilized with mol% of CeO\({}_{2}\) evidenced in the different experimental studies and this paper (the \(X\)-axis shows the mol%, while the \(Y\)-axis shows the number of mentions of these structures). Figure 7: SEM image of a fracture surface of ZrO\({}_{2}\)–5 wt.% CeO\({}_{2}\) sintered sample. The destruction of grains observed in Figure 6 is predominantly transcrystalline, which indicates the strength of inter-grain boundaries. The bending strength \(\sigma_{b}\) of the specimen under three-point bending conditions was determined according to the following formula:  
\[\sigma_{b}=\frac{3 Fl}{2 bh^{2}} \tag{1}\]  
where \(F\) is the force value at the moment of separation of the sample into parts, N; \(l\) is the length of the sample, mm; \(b\) is the width of the sample, mm; \(h\) is the thickness of the sample in the parallel direction to the force applied to the sample, mm. Different brittle materials experience rupture at different maximum tensile stresses. Statistically, the brittle fracture is dependent on the weakest element in the material's volume. According to the two-parameter Weibull distribution, the probability of failure \(P_{f}\) at a given the stress value, \(\sigma\), is defined as follows [42; 43]:  
\[P_{f}=1-\exp\left[-\left(\frac{\sigma}{\sigma_{0}}\right)^{m}\right] \tag{2}\]  
where \(m\) is the shape parameter of the Weibull distribution; \(\sigma\) is the actual inner stress under the applied force, MPa; and \(\sigma_{0}\) denotes the scale parameter, MPa. (The threshold value, below which no specimen is expected to fail, is neglected). To estimate the the Weibull parameters, Equation (1) is twice logarithm and reduced to a linear form, as follows:  
\[\ln\left(\ln\left(1-P_{f}\right)^{-1}\right)=m\ln(\sigma)-m\ln(\sigma_{0}). \tag{3}\]  
The shape and scale parameters, \(m\) and \(\sigma_{0}\), respectively, are obtained from Equation (3) by the least squares method. The density of fracture probability \(f\) dependent on the value of \(\sigma_{0}\) is described by the equation, as follows:  
\[f(\sigma)=\frac{m}{\sigma_{0}}\left(\frac{\sigma}{\sigma_{0}}\right)^{m-1} \exp\left[-\left(\frac{\sigma}{\sigma_{0}}\right)^{m}\right] \tag{4}\]  
To perform the calculations, the entered experimental data are ordered in ascending order, and for each \(i\)-th value, the empirical probability function \(P_{exp}(i)\) is determined according to the following formula:  
\[P_{exp}(i)=\frac{i-0.5}{N}, \tag{5}\]  
where \(N\) is the number of tested samples. The results of the tensile strength measurement and of related calculations are collected in Table 2. \begin{table}
\begin{tabular}{c c c} \hline
**Experiment No.** & \(\sigma_{f}\)**MPa** & **ln(ln(1 - \(P_{f}\))\({}^{-1}\))** \\ \hline
1 & 370 \(\pm\) 3 & –3.218 \\
2 & 390 \(\pm\) 3 & –2.355 \\
3 & 400 \(\pm\) 3 & –2.068 \\
4 & 420 \(\pm\) 5 & –1.349 \\
5 & 430 \(\pm\) 5 & –1.061 \\
6 & 440 \(\pm\) 5 & –0.630 \\
7 & 446 \(\pm\) 5 & –0.486 \\ \hline \end{tabular}
\end{table} 
Table 2: Bending strength \(\sigma=\sigma_{f}\) and the probability of failure \(P_{f}\) for the sintered samples of ZrO\({}_{2}\)–5 wt.% CeO\({}_{2}\). It is noteworthy that the high Pearson's coefficient 0.999 and low curve fit standard deviation s = 0.28 of the experimental points from linearity suggest the high quality fit of the Weibull distribution. The slope of the line is the shape parameter \(m\) of the Weibull distribution, the so-called modulus. Its positive value characterizes the curve as S-shaped with upward curvature followed by a turning (inflection) point. The m value of 14.383 indicates the variability of the material's strength. A relatively high Weibull modulus suggests a narrow flaw size population and, in a consequence, a narrow range of bending strengths.