Theoretical approach to determine dynamic fatigue strength characteristics of ceramics under variable loading rates on the basis of SCG concept

Abstract

This paper presents a theoretical approach to determine the dynamic fatigue strength characteristics of ceramics under variable loading rates on the basis of the slow crack growth (SCG) concept. First, a probabilistic effective inert strength model was derived on the basis of the SCG concept in conjunction with the Weibull distribution for ceramics subjected to multi-stage loading. Second, a four-point bending test was conducted on $\hbox {Al}_{{2}}\hbox {O}_{{3}}$ under constant and two-stage variable loading rates, and the fracture surface was then observed. The experimental data that depend on loading rates can be unifiedly evaluated after converting the data to the effective inert strength, obeying the three-parameter Weibull distribution. In addition, the Weibull plots of the inert strength, which were calculated from the inclusion size on the fracture surface using the grain fracture model, showed good agreement with the three-parameter Weibull distribution for the converted effective inert strength. These analytical results theoretically indicate that dynamic fatigue under variable loading rates occurs by obeying SCG at the inclusion. Further, the inert strength and its scatter depend on the size and distribution of inclusions.

Appendix: Determination of SCG parameters

The SCG parameters n and $g_{\mathrm{o}}$ were determined using the $\sigma _{\mathrm {max}} -\dot{\sigma }$ plots (loading rate $\dot{\sigma })$ in the CLR test shown in Fig. 6a by rewriting Eq. (9) as follows:

$$\begin{aligned} y=\ln \left( {\frac{S_{\mathrm{i}}^*}{\sigma _{\mathrm {max}} }} \right) =\frac{1}{n-2}\ln \left( {1+g_{\mathrm{o}} \sigma _{\mathrm {max}} ^{2}t_{\mathrm {eff}} } \right) . \end{aligned}$$

(18)

The ratio $S_{\mathrm{i}}^*/\sigma _{\mathrm {max}}$ on the left-hand side of Eq. (18) was plotted against $x=\sigma _{\mathrm {max}} ^{2}t_{\mathrm {eff}}$. Here, the effective inert strength $S_{\mathrm{i}}^*$ was assumed to be the average of $\sigma _{\mathrm {max}}$ in the CLR test under $\hbox {CHS} = 200\,\hbox {mm}\,\hbox {min}^{-1}$ , which converges to the inert strength as shown in Fig. 6a, and the effective loading time $t_{\mathrm {eff}}$ was calculated using Eq. (4) and the applied stress $\sigma \left( t \right) =\dot{\sigma }t$ (period stress $\sigma _{\mathrm {max}} =\dot{\sigma }t_{\mathrm{f}} )$ as $\sigma _{\mathrm {max}}/\left\{ {\dot{\sigma }\left( {n+1} \right) } \right\}$. Then, n and $g_{\mathrm{o}}$ were obtained by curve fitting these plots to the equation $y=1/\left( {n-2} \right) \ln \left( {1+g_{\mathrm{o}} x} \right)$ using the least squares method, as shown in Fig. 6b. The values of n and $g_{\mathrm{o}}$ were obtained as 21.1 and $2.54\times 10^{-3}\,\hbox {MPa}^{-2}\,\hbox {s}^{-1}$, respectively. The value of n was smaller than $n = 37.5$, which was obtained by Ritter and Humenik (1979). They conducted the CLR test under the loading rate region from 1/10 to 1/100 of the lowest loading rate of the CLR test in this study. Then, n was analyzed by curve fitting the obtained $\sigma _{\mathrm {max}} -\dot{\sigma }$ plots to the equation $\sigma _{\mathrm {max}} \propto \dot{\sigma }^{1/\left( {n+1} \right) }$, which is obtained on the basis of the SCG concept. On the other hand, on analyzing the S–N diagram in the cyclic fatigue test and the S–t diagram in the static fatigue test, the value of n varied between 21–25 and 36–54, respectively (Guiu et al. 1991). The value of n obtained from the S–N diagram almost agrees with this experimental result. The value of n differs depending on the loading method and the loading rate region.