Mechanics-based statistics of failure risk of quasibrittle structures and size effect on safety factors

Abstract

In mechanical design as well as protection from various natural hazards, one must ensure an extremely low failure probability such as 10 ؊6 . How to achieve that goal is adequately understood only for the limiting cases of brittle or ductile structures. Here we present a theory to do that for the transitional class of quasibrittle structures, having brittle constituents and characterized by nonnegligible size of material inhomogeneities. We show that the probability distribution of strength of the representative volume element of material is governed by the Maxwell-Boltzmann distribution of atomic energies and the stress dependence of activation energy barriers; that it is statistically modeled by a hierarchy of series and parallel couplings; and that it consists of a broad Gaussian core having a grafted far-left power-law tail with zero threshold and amplitude depending on temperature and load duration. With increasing structure size, the Gaussian core shrinks and Weibull tail expands according to the weakest-link model for a finite chain of representative volume elements. The model captures experimentally observed deviations of the strength distribution from Weibull distribution and of the mean strength scaling law from a power law. These deviations can be exploited for verification and calibration. The proposed theory will increase the safety of concrete structures, composite parts of aircraft or ships, microelectronic components, microelectromechanical systems, prosthetic devices, etc. It also will improve protection against hazards such as landslides, avalanches, ice breaks, and rock or soil failures.cohesive fracture ͉ extreme value statistics ͉ activation energy ͉ Maxwell-Boltzmann ͉ scaling

Scaling of strength and lifetime probability distributions of quasibrittle structures based on atomistic fracture mechanics

“…5). Isn't that in conflict with the preceding simpler model (1), in which the elements of the hierarchical model had the tail exponent of 1? No, because the elements in ref.…”

Measurement of the Mechanical Properties of Bone: A Recent History

“…Therefore, the lower the value of m the more important the size effect is likely to be. There are other ways in which a volume effect may be produced, apart from the ''weakest link'' described above, possibly particularly relevant for semibrittle materials like bone, and interested readers should read Bažant [3] or Bažant and Pang [4].…”

Size effect on strength and lifetime probability distributions of quasibrittle structures

“…Though various stochastic multiscale numerical approaches have been proposed (Graham-Brady et al 2006;Williams & Baxer 2006;Xu 2007), the capability of these approaches is always limited due to incomplete knowledge of the uncertainties in the information across all the scales. Instead, for the sole purpose of statistics, the multi-scale transition of strength statistics has been characterized by a hierarchical model, which consists of bundles and chains shown in figure 1 (Bažant & Pang 2006, 2007.…”

“…The transition of strength statistics from the nano-scale to the RVE scale can be mechanically represented by a hierarchical model consisting of bundles and chains (Bažant & Pang 2006, 2007. Based on the asymptotic properties of strength cdfs of bundles and chains, it has been shown that the strength cdf of one RVE can be approximately modelled as a Gaussian distribution onto which a power-law tail is grafted at the failure probability of about P f ≈ 10 −4 -10 −3 .…”

“…One must consider a finite weakest-link model because of the fact that the size of one RVE is not negligible compared to the structure size, which is the salient feature of all quasibrittle structures. A probabilistic theory was recently developed to model the strength distribution of quasibrittle structures (Bažant & Pang 2006, 2007.…”