Statistics of strength of ceramics: finite weakest-link model and necessity of zero threshold
It is expected that the distribution of strength of quasibrittle structures changes from predominantly Gaussian to Weibullian distribution as the structure size increases. Therefore, the ratio of the safety factor to the standard deviation is nearly double when passing from small laboratory samples to quasibrittle structures much larger than the representative volume element (RVE). The present paper provides a verification of such behavior for quasibrittle ceramics. The developed finite weakest-link model is based on the fact that the rate processes (e.g. effects of temperature and activation energy) are not the only phenomena that translate from the nanoscale to the macroscale material. The tail of the cumulative distribution function (cdf) of failure probability versus strength is another such phenomenon, namely due to (i) the left tail of cdf of strength on the atomic scale is described by a power law with a zero threshold, and (ii) this power law with zero threshold is indestructible, and its exponent gets raised when passing to higher scales. The paper explores these two properties for the cdf of strength for structures consisting of touch ceramics by considering the structures of positive geometry. In this case, the derivative of stress intensity factor with respect to crack length at constant load is positive, causing failure occurring at macrocrack initiation from RVE. According to the theory, the Weibull distribution corresponds to the weakest-link chain model in which the number of links tends to infinity. When the specimen is not large enough compared to the grain size, a chain with a finite number of links must be used that leads to a distribution that differs from Weibull one. The proposed model agrees with the measured mean size effect curves for different ceramics. It is justified by energy release rate dependence of the activation energy barriers for random crack length jumps through the atomic lattice. This shows that the tail of the failure probability distribution should be a power law with zero threshold. It is also shown that the coefficient of strength variation may decrease significantly with the structure size at the quasibrittle behavior. This opposite effect may completely offset the effect of the relative distance increased from the mean.
scientific article; zbMATH DE number 5605034