Question 7 of 8: CaO Coordination Number; Porosity, Grain Size, and Weibull Statistics in Ceramics

Question 7: CaO Coordination Number; Porosity, Grain Size, and Weibull Statistics in Ceramics (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. $r_{Ca^{2+}}=0.100$ nm, $r_{O^{2-}}=0.140$ nm.

Find. (a) Coordination number of CaO. (b) Porosity and grain-size effects on ceramic tensile strength. (c) Why statistical (Weibull) methods are used for ceramic failure prediction.

Approach

Part (a) applies the radius-ratio rule directly. Parts (b) and (c) are conceptual: porosity and grain size are treated as flaw-population parameters that control the size of the critical crack a ceramic can tolerate before brittle fracture, and the Weibull distribution is introduced as the standard statistical framework for a strength that is controlled by the largest flaw in a random population rather than by a single deterministic material property.

1. (a) Radius ratio. $$\frac{r_{cation}}{r_{anion}}=\frac{r_{Ca^{2+}}}{r_{O^{2-}}}=\frac{0.100}{0.140}=\boxed{0.714}.$$ This falls in the range $0.414\text{–}0.732$, which predicts octahedral coordination, CN $=\boxed{6}$ — consistent with CaO's actual rock-salt (NaCl-type) crystal structure.
2. (b) Porosity effect. Pores act as pre-existing stress-concentrating flaws and also directly reduce the load-bearing cross-sectional area, so tensile strength decreases with increasing volume fraction porosity $P$, commonly modeled as an exponential decay $\sigma=\sigma_0\exp(-nP)$; even a few percent porosity can substantially cut strength because strength is controlled by the largest/sharpest pore, not the average pore size.
3. Grain size effect. Finer grain size increases tensile strength (a Hall-Petch-type relationship, $\sigma\propto d^{-1/2}$), because grain boundaries impede crack propagation and a fine-grained structure statistically limits the maximum flaw size that can exist within any one grain or grain facet.
4. (c) Why statistical (Weibull) methods. Ceramics fail in a brittle manner, with essentially no plastic deformation to blunt a crack tip or redistribute stress away from a flaw; fracture strength is therefore controlled by the single most severe flaw in the stressed volume, and flaw size/severity varies randomly from specimen to specimen (and even from region to region within one specimen). This makes ceramic strength an inherently statistical, not deterministic, property — unlike a ductile metal's yield strength, which is comparatively insensitive to any one flaw.
5. Weibull distribution. The Weibull distribution models the probability of survival at stress $\sigma$ for a component of volume $V$ as $$P_{surv}=\exp\!\left[-\frac{V}{V_0}\left(\frac{\sigma}{\sigma_0}\right)^m\right],$$ where $m$, the Weibull modulus, is a shape parameter describing scatter (a large $m$ means tightly-clustered strengths / a narrow flaw-size distribution; a small $m$ means widely scattered strengths). Because $P_{surv}$ depends on volume $V$, the distribution also predicts (and correctly captures) the observed size effect in ceramics: larger specimens are statistically more likely to contain a critical flaw and therefore test weaker, on average, than smaller specimens of the identical material.