Scaling strength distributions in quasi-brittle materials from micro- to macro-scales: A computational approach to modeling Nature-inspired structural ceramics

• Analysis of quasi-brittle failure including statistical and size-dependent aspects.
• Use of computational homogenization to compute up to macroscopic scale samples.
• Application to Nature-inspired ceramic structures made by freeze-casting.
• Strength distribution shape converges to Weibull for macroscopic scale samples.
• Weakest-link scaling does not apply to the Weibull-like macroscopic strength distributions.

This paper presents an approach to predict the strength distribution of quasi-brittle materials across multiple length-scales, with emphasis on Nature-inspired ceramic structures. It permits the computation of the failure probability of any structure under any mechanical load, solely based on considerations of the microstructure and its failure properties by naturally incorporating the statistical and size-dependent aspects of failure. We overcome the intrinsic limitations of single periodic unit-based approaches by computing the successive failures of the material components and associated stress redistributions on arbitrary numbers of periodic units. For large size samples, the microscopic cells are replaced by a homogenized continuum with equivalent stochastic and damaged constitutive behavior. After establishing the predictive capabilities of the method, and illustrating its potential relevance to several engineering problems, we employ it in the study of the shape and scaling of strength distributions across differing length-scales for a particular quasi-brittle system. We find that the strength distributions display a Weibull form for samples of size approaching the periodic unit; however, these distributions become closer to normal with further increase in sample size before finally reverting to a Weibull form for macroscopic sized samples. In terms of scaling, we find that the weakest link scaling applies only to microscopic, and not macroscopic scale, samples. These findings are discussed in relation to failure patterns computed at different size-scales.