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In shear failure, reinforced concrete (RC) beams always develop, in a stable manner, a finite length crack before the maximum load is reached. Thus, the crack tip location cannot sample a large volume of material with random strength because a small region in which the crack tip can lie is fixed by fracture mechanics. Consequently, the size effect on the mean strength cannot be statistical. It must be predominantly energetic or deterministic and, thus, must follow the Type-2 size effect law. What has not yet been clarified is the size effect on the coefficient of variation (CoV) of beam strength, which is important for anchoring the probability distribution of shear strength and choosing the safety factor. In this study, we run thousands of explicit finite element simulations using Abaqus-Explicit version 6.14 with microplane model M7, each with a random input of material strength and Young's modulus for each finite element in the structure. The CoV of beam strength is found to decrease with the structure size when geometrically similar beams are compared, although the CoV tends to a constant for large sizes. This size effect on the CoV is similar to that in ductile failure governed by a Gaussian distribution of strength and contrasts with that in brittle failures following the Weibull distribution, for which the CoV is size independent. To characterize the size dependence of the strength CoV, an analytical formula is developed based on the statistics of the sample quantiles of a series of random variables. [ABSTRACT FROM AUTHOR]

Highlights: [•] We present an indirect method to determine the strength distribution of quasibrittle structures. [•] The method is verified by a set of experiments on asphalt mixture at a low temperature. [•] The finite weakest link model agrees well with the measured strength histograms of specimens of different sizes. [ABSTRACT FROM AUTHOR]

Engineering structures such as aircraft, bridges, dams, nuclear containments and ships, as well as computer circuits, chips and MEMS, should be designed for failure probability < 10-10 per lifetime. The safety factors required to ensure it are still determined empirically, even though they represent much larger and much more uncertain corrections to deterministic calculations than do the typical errors of modern computer analysis of structures. The empirical approach is sufficient for perfectly brittle and perfectly ductile structures since the cumulative distribution function (cdf) of random strength is known, making it possible to extrapolate to the tail from the mean and variance. However, the empirical approach does not apply to structures consisting of quasibrittle materials, which are brittle materials with inhomogeneities that are not negligible compared to structure size. This paper presents a refined theory on the strength distribution of quasibrittle structures, which is based on the fracture mechanics of nanocracks propagating by activation energy controlled small jumps through the atomic lattice and an analytical model for the multi-scale transition of strength statistics. Based on the power law for creep crack growth rate and the cdf of material strength, the lifetime distribution of quasibrittle structures under constant load is derived. Both the strength and lifetime cdf's are shown to be size- and geometry-dependent. The theory predicts intricate size effects on both the mean structural strength and lifetime, the latter being much stronger. The theory is shown to match the experimentally observed systematic deviations of strength and lifetime histograms of industrial ceramics from the Weibull distribution. [ABSTRACT FROM AUTHOR]

This paper presents a general size effect equation for the strength of hybrid structures, which are made of two dissimilar quasibrittle materials with a thin and weak bimaterial interface. Depending on the material mismatch and structure geometry, a singular stress field could occur at the bimaterial corner. For structures with strong stress singularities, an energetic size effect is derived based on the equivalent linear elastic fracture mechanics and asymptotic matching. For structures without stress singularities, a finite weakest link model is adopted to derive the size effect. A general scaling equation that bridges the limits of strong and zero stress singularities is formulated by combining the energetic scaling of fracture of the bimaterial corner and the finite weakest link model. [ABSTRACT FROM AUTHOR]

Abstract: The statistics of structural lifetime under constant load are related to the statistics of structural strength. The safety factors applied to structural strength must ensure failure probability no larger than , which is beyond the means of direct verification by histogram testing. For perfectly brittle materials, extrapolation from the mean and variance to such a small tail probability is no problem because it is known that the Weibull distribution applies. Unfortunately, this is not possible for quasibrittle materials because the type of cumulative distribution function (cdf) has been shown to vary with structure size and shape. These are materials with inhomogeneities and fracture process zones (FPZ) that are not negligible compared to structural dimensions. A probabilistic theory of strength of quasibrittle structures failing at macro-crack initiation, which can be experimentally verified and calibrated indirectly, has recently been deduced from the rate of jumps of atomic lattice cracks governed by activation energy barriers. This paper extends this nano-mechanics based theory to the distribution of structural lifetime. Based on the cdf of strength and a power law for subcritical crack growth rate, the lifetime cdf of quasibrittle structures under constant loads is derived. The lifetime cdf is shown to depend strongly on the structure size as well as geometry. It is found that, for the creep rupture case, the mean structural lifetime exhibits a very strong size effect, much stronger than the size effect on the mean structure strength. The theory also implies temperature dependence of the lifetime cdf. For various quasibrittle materials, such as industrial ceramics and fiber composites, it is demonstrated that the proposed theory correctly predicts the experimentally observed deviations of lifetime histograms from the Weibull distribution. [Copyright &y& Elsevier]

Owing to the random nature of heterogeneity, damage and fracture behavior of quasi-brittle materials exhibits a considerable degree of uncertainty. Computational modeling of stochastic fracture in quasi-brittle materials has become an indispensable tool for analysis and design of engineering structures. To this end, we present in this paper a computational framework to capture probabilistic fracture in heterogeneous quasi-brittle solids by combining the random field theory and the phase-field cohesive zone model (PF-CZM). The spatial variation of the material strength and fracture energy is represented by a cross-correlated bivariate random field generated by the Karhunen–Loève expansion. The recently proposed PF-CZM is employed to simulate the stochastic crack nucleation and propagation in quasi-brittle solids. The objectivity of the Monte-Carlo simulation is achieved by imposing a specific condition on the phase-field length scale parameter and the correlation length of the random field. In particular, upon this condition the width of the fracture process zone (FPZ) is considerably smaller than the correlation length of the random field such that the material inside the FPZ does not exhibit significant spatial variations of mechanical properties. As the fracture energy is intrinsically incorporated in the PF-CZM, it is unnecessary in this case to explicitly consider the FPZ width. The resulting probabilistic PF-CZM together is applied to the Monte-Carlo simulations of fracture in concrete structures of different geometries. It is shown that the stochastic simulation results are insensitive to both the phase-field length scale parameter and the finite element mesh discretization as in the previous deterministic analyses. Enhanced with the specific condition on the involved characteristic lengths, the PF-CZM provides a viable tool for stochastic simulations of damage and fracture in quasi-brittle structures. [ABSTRACT FROM AUTHOR]