Scaling theory for quasibrittle structural failure

Abstract

This inaugural article has a twofold purpose: (i) to present a simpler and more general justification of the fundamental scaling laws of quasibrittle fracture, bridging the asymptotic behaviors of plasticity, linear elastic fracture mechanics, and Weibull statistical theory of brittle failure, and (ii) to give a broad but succinct overview of various applications and ramifications covering many fields, many kinds of quasibrittle materials, and many scales (from 10 ؊8 to 10 6 m). The justification rests on developing a method to combine dimensional analysis of cohesive fracture with second-order accurate asymptotic matching. This method exploits the recently established general asymptotic properties of the cohesive crack model and nonlocal Weibull statistical model. The key idea is to select the dimensionless variables in such a way that, in each asymptotic case, all of them vanish except one. The minimal nature of the hypotheses made explains the surprisingly broad applicability of the scaling laws. Discovery of the concept of stress and strength by Galileo (1) may, in retrospect, be regarded as the first scaling theory of solid mechanics. This theory, now known to apply only to elastoplastic behavior, captures the fact that, under controlled load P, geometrically similar structures of different sizes D fail at the same nominal stress N , defined as the maximum stress in the structure (if no stress singularities exist) or simply as N ϭ P͞D 2 or P͞bD for three-or two-dimensional scaling (b ϭ structure thickness). Any departure from such scaling came to be known as the ''size effect.'' About 350 years ago, Mariotte (2) pointed out that a size effect must arise because the local material strength is random and its minimum encountered in a structure decreases with D. Nevertheless, proper mathematical formulation of this idea had to wait until 1939. That year, Weibull (3) experimentally demonstrated that N for brittle structures has the probability distribution that came to bear his name, although already in 1928 this distribution was derived by Fischer and Tippett (4) as the only possible limiting distribution with a threshold for the minimum of a set of n independent random variables for n 3 ϱ. The tail of this distribution is a power law, from which Weibull deduced that the statistical size effect is also a power law, the exponent of which is a function of the coefficient of variation of material strength.For half a century afterward, whenever a size effect was observed, it was generally attributed to Weibull theory (3, 5, 6), which was amply confirmed for fatigued metals and fine-grained ceramics. However, beginning with studies of concrete for nuclear reactors during 1970-1985, it gradually transpired that Weibull theory does not apply to materials now termed ''quasibrittle'' (7-9). These are materials in which the fracture process zone (FPZ) is not negligible compared with the cross-section dimension D (and may even encompass the entire cross section).Depending on the scale of observation or application, quasib...

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