Statistics of strength of ceramics: finite weakest-link model and necessity of zero threshold

Abstract

It is argued that, in probabilistic estimates of quasibrittle structure strength, the strength threshold should be considered to be zero and the distribution to be transitional between Gaussian and Weibullian. The strength histograms recently measured on tough ceramics and other quasibrittle materials, which have been thought to imply a Weibull distribution with nonzero threshold, are shown to be fitted equally well or better by a new weakest-link model with a zero strength threshold and with a finite, rather than infinite, number of links in the chain, each link corresponding to one representative volume element (RVE) of a non-negligible size. The new model agrees with the measured mean size effect curves. It is justified by energy release rate dependence of the activation energy barriers for random crack length jumps through the atomic lattice, which shows that the tail of the failure probability distribution should be a power law with zero threshold. The scales from nano to macro are bridged by a hierarchical model with parallel and series couplings. This scale bridging indicates that the power-law tail with zero threshold is indestructible while its exponent gets increased on each passage to a higher scales. On the structural scale, the strength distribution except for its far left power-law tail, varies from Gaussian to Weibullian as the structure size increases. For the mean structural strength, the theory predicts a size effect which approaches the Weibull power law asymptotically for large sizes but deviates from it at small sizes. This deviation is the easiest way to calibrate the theory experimentally. The structure size is measured in terms of the number of RVEs. This number must be convoluted by an integral over the dimensionless stress field, which depends on structure geometry. The theory applies to the broad class of structure geometries for which failure occurs at macro-crack initiation from one RVE, but not to structure geometries for which stability is lost only after large macro-crack growth. Based on tolerable structural failure probability of <10−6, the change from nonzero to zero threshold may often require a major correction in safety factors.