The Breaking of Brittle Materials

Abstract

We present a unified theory of fracture in disordered brittle media that reconciles apparently conflicting results reported in the literature. Our renormalization group based approach yields a phase diagram in which the percolation fixed point, expected for infinite disorder, is unstable for finite disorder and flows to a zero-disorder nucleation-type fixed point, thus showing that fracture has a mixed first order and continuous character. In a region of intermediate disorder and finite system sizes, we predict a crossover with mean-field avalanche scaling. We discuss intriguing connections to other phenomena where critical scaling is only observed in finite size systems and disappears in the thermodynamic limit. DOI: 10.1103/PhysRevLett.110.185505 PACS numbers: 62.20.mj, 62.20.mm, 62.20.mt, 64.60.ae Brittle fracture in disordered media intertwines two phenomena that seldom coexist, namely, nucleation and critical fluctuations. The usual dichotomy of thought between nucleated and continuous transitions makes the study of fracture interesting. Even more intriguing is the fact that crack nucleation happens at zero stress in the thermodynamic limit: smaller is stronger and larger is weaker. This makes the existence of critical fluctuations in the form of clusters and avalanches of all sizes even more mysterious. What kind of critical point governs a phase transition that happens at zero applied field (stress) in the thermodynamic limit, and what is the universality class of such a transition? How do self-similar clusters, extremely rough crack surfaces, and scale invariant avalanches ultimately give rise to sharp cracks and localized growth? These questions have been addressed previously via a host of different theories, such as those based on percolation and multifractals [1][2][3][4], spinodal modes and meanfield criticality [5], and classical nucleation [6][7][8][9]. In this Letter, we present a theoretical framework based on the renormalization group and crossover scaling that unifies the seemingly disparate descriptions of fracture into one consistent framework.Fracture in disordered media is the result of a complex interplay between quenched heterogeneities and long-range stress fields leading to diffuse damage throughout the sample, and local stress concentration favoring the formation of sharp localized cracks. The self-affine morphology of cracks [10], the power-law statistics of avalanche precursors [11][12][13][14], and the scale dependence of the failure strength distribution [15][16][17] all result from this competition. Disordered fracture can be understood in the limit of infinitesimal as well as infinite disorder. Infinitesimal disorder means a perfect crystalline material with just a few isolated defects (say a missing atom or a microcrack). In this limit, fracture statistics can be understood as a nucleation-type first order phase transition [6][7][8][9]. In the limit of infinite disorder, stress concentration becomes irrelevant and fracture progresses via uncorrelated percolationlike damage [...

Grand Challenges in Glass Science

“…High-strength glasses are available (Varshneya, 2010;Wondraczek et al, 2011); however, even with chemical strengthening, they are still below the theoretical strength aimed for in defect-free conditions (Mattos, 2013). To approach the theoretical limit, understanding crack formation (Wiederhorn et al, 2013) and fracture behavior (Quinn, 2012;Bouchaud, 2013) is necessary. Further research should also aim to improve the ductility and fracture toughness of glass (Shi et al, 2014), which are intimately connected to the Poisson's ratio of the material (Greaves et al, 2011).…”

Fatigue behavior of HPC and FRC under cyclic tensile loading: Experiments and modeling

“…Illustration of the fiber‐microcrack interaction using a micromechanics model . From left to right: Representative elementary volume, single fiber (black) bridging a microcrack (white), slip strain S(y) at the fiber‐matrix interface, stress equilibrium at fiber scale…”

“…While a fracture mechanics based analysis can help in the design of fatigue tolerant high performance FRC, it does not provide insight into the role of disorder on the microscale level of heterogeneous materials such as concrete. Introducing disorder into a homogeneous material has important consequences on the stress that the material can withstand before loss of material integrity . One approach to account for the mechanics of disorder is to introduce a certain strength distribution into a homogeneous solid.…”