Issue 24

M. Davydova et alii, Frattura ed Integrità Strutturale, 24 (2013) 60-68; DOI: 10.3221/IGF-ESIS.24.05 60 Special Issue: Russian Fracture Mechanics School Fractal statistics of brittle fragmentation M. Davydova, S. Uvarov Institute of Continuous Media Mechanics Ural Branch Russian Academy of Sciences, 1, Ac. Korolev str., 614013 Perm, Russia. davydova@icmm.ru ; usv@icmm.ru A BSTRACT . The study of fragmentation statistics of brittle materials that includes four types of experiments is presented. Data processing of the fragmentation of glass plates under quasi-static loading and the fragmentation of quartz cylindrical rods under dynamic loading shows that the size distribution of fragments (spatial quantity) is fractal and can be described by a power law. The original experimental technique allows us to measure, apart from the spatial quantity, the temporal quantity - the size of time interval between the impulses of the light reflected from the newly created surfaces. The analysis of distributions of spatial (fragment size) and temporal (time interval) quantities provides evidence of obeying scaling laws, which suggests the possibility of self- organized criticality in fragmentation. K EYWORDS . Fragmentation of brittle materials; Fractal statistics; Self-organized criticality. I NTRODUCTION ragmentation is the process of breaking a solid into separate fragments caused by multiple fractures. Such phenomenon can be observed in both engineering and natural objects over a wide range of spatial and temporal scales. An investigation of fragmentation statistics generally includes the determination of the cumulative distribution of fragment sizes or masses, i.e., the number of fragments ( ) N m with a size or mass larger than S or m , respectively. The distribution type depends on loading conditions, material characteristics and sample geometries. Many types of distribution functions have been observed experimentally: log-normal, power-law, Mott, exponential, Weibull, and combined exponential and power-law [1 - 8]. Summarizing the results of experimental data processing, we can classify all these distribution functions into two groups: exponential and power law. The assumption that the exponential distribution is typical of the fragmentation of ductile materials and the power-law distribution characterizes brittle fragmentation has been discussed by Grady [7]. Donald Turcotte [8] has pursued the fragmentation of brittle materials as a fractal process resulting in the power law distribution function d N x   , where N is the number of fragments, x is the linear dimension of fragments, and d is the fractal dimension. The fractal character of the distribution function in a wide range of fragment sizes allows Oddershede et al. [3] to suppose that the fragmentation exhibits self-organized criticality (SOC). In their seminal paper Per Bak, Chao Tang and Kurt Wiesenfeld [9] present a new concept of SOC; numerical simulation allows them to describe the behavior of a sand pile (example of a self-organized critical system) and to conclude:  distribution of life times of avalanches obeys a power law;  distribution of avalanche sizes follows a power law as well. In other words, to prove that the system exhibits SOC, it is necessary to establish the existence of a power law for temporal and spatial quantities. The purpose of this experimental study is to demonstrate that the fragmentation process exhibits SOC. To this end, we need to determine the distribution of temporal and spatial variables. It has been shown that for the fragmentation of glass plates under quasi-static loading there exists scaling in the crack pattern formed during fragmentation and fragment size distribution are also described by the power law. The experiments on quartz cylindrical F