(Finite) statistical size effects on compressive strength

The larger structures are, the lower their mechanical strength. Already discussed by Leonardo da Vinci and Edmé Mariotte several centuries ago, size effects on strength remain of crucial importance in modern engineering for the elaboration of safety regulations in structural design or the extrapolation of laboratory results to geophysical field scales. Under tensile loading, statistical size effects are traditionally modeled with a weakest-link approach. One of its prominent results is a prediction of vanishing strength at large scales that can be quantified in the framework of extreme value statistics. Despite a frequent use outside its range of validity, this approach remains the dominant tool in the field of statistical size effects. Here we focus on compressive failure, which concerns a wide range of geophysical and geotechnical situations. We show on historical and recent experimental data that weakest-link predictions are not obeyed. In particular, the mechanical strength saturates at a nonzero value toward large scales. Accounting explicitly for the elastic interactions between defects during the damage process, we build a formal analogy of compressive failure with the depinning transition of an elastic manifold. This critical transition interpretation naturally entails finite-size scaling laws for the mean strength and its associated variability. Theoretical predictions are in remarkable agreement with measurements reported for various materials such as rocks, ice, coal, or concrete. This formalism, which can also be extended to the flowing instability of granular media under multiaxial compression, has important practical consequences for future design rules.Owing to its importance for structural design (1), the elaboration of safety regulations (2), or the extrapolation of laboratory results to geophysical field scales (3), the size effects on strength of materials are one of the oldest problems in engineering, already discussed by Leonardo da Vinci and Edmé Mariotte (4) several centuries ago, but still an active field of research (5, 6). As early as 1686, Mariotte (4) qualitatively introduced the weakest-link concept to account for size effects on mechanical strength, a phenomenon evidenced by Leonardo da Vinci almost two centuries earlier. This idea, which states that the larger the system considered is, the larger the probability to find a particularly faulty place that will be at the origin of global failure, was formalized much later by Weibull (7). Considering a chain of elementary independent links, the failure of the chain is obtained as soon as one link happens to break. By virtue of the independence between the potential breaking events, the survival probability of a chain of N links is obtained by the simple multiplication of the N elementary probabilities. Depending on the properties of the latter, the global survival probability converges toward one of the three limit distributions identified by Weibull (7), Gumbel (8), and Fréchet (8), respectively. Together with Fisher and Tippett (9), these authors pioneered the field of extreme value statistics.This purely statistical argument, undoubtedly valid in 1D, was extended by Weibull (7, 10) to account for the risk of failure of 3D samples or structures. Besides the hypothesis of independence, it thus requires an additional hypothesis of brittleness: The nucleation of any elementary crack at the microscopic scale from a preexisting flaw is assumed to immediately induce the failure at the macroscale. More specifically, following linear elastic fracture mechanics (LEFM) stating that crack initiation from a flaw of size s occurs at a stress , one gets a probability of failure of a system of size L under an applied stress σ, , that depends on the distribution of preexisting defect sizes. Assuming a power law tail for this distribution, Weibull statistics are expected(7), , whereas Gumbel statistics are expected for any distribution of defect sizes whose the tail falls faster than that of a power law (8, 11, 12), , where m is the so-called Weibull’s modulus, d is the topological dimension, and L₀ and σ_u are normalizing constants. For Weibull statistics, the mean strength and the associated SD δ(σ_f) then scale with sample size L as . This approach has been successfully applied to the statistics of brittle failure strength under tension (7, 13), with m in the range 6–30 (14). It implies a vanishing strength for L → +∞, although this decrease can be rather shallow, owing to the large values of m often reported.Although relying on strong hypotheses, this weakest-link statistical approach was almost systematically invoked until the 1970s to account for size effects on strength whatever the material and/or the loading conditions. However, as shown by Bazant (1, 5), in many situations the hypothesis of brittleness is not obeyed. This is in particular the case when the size of the fracture process zone (FPZ) becomes nonnegligible with respect to the system size. In this so-called quasi-brittle case, an energetic, nonstatistical size effect applies (15), which has been shown to account for a large variety of situations (5). Toward large scales, i.e., L → +∞, the FPZ becomes negligible compared with L, and the hypothesis of brittleness should therefore be recovered and statistical size effects should dominate. Statistical numerical models of fracture of heterogeneous media also revealed deviations from the extreme value statistics predictions (16) but, as stated by Alava et al. (ref. 11, p. 9), “the role of damage accumulation for fracture size effects in unnotched samples still remains unclear.” As shown below, compressive failure results from such progressive damage accumulation.In what follows, we do not consider (deterministic) energetic size effects and explore a situation, compressive failure, where both the hypotheses of brittleness (in the sense given above) and independence are not fulfilled, up to very large scales. Relaxing these initial hypotheses of the weakest-link theory, our statistical physics approach remains statistical by nature and relies on the interplay between internal disorder and stress redistributions. It is based on a mapping of brittle compressive failure onto the critical depinning transition of an elastic manifold, a class of models widely used in nonequilibrium statistical physics characterized by a dynamic phase transition (17). This approach does not consider a sample’s shape effects (18), only statistical size effects. The critical scaling laws associated to this phase transition naturally predict a saturation of the compressive strength at a large scale and are in remarkable agreement with measurements reported for various materials such as rocks, ice, coal, or concrete.