Unified physics of stretched exponential relaxation and Weibull fracture statistics

Abstract

The complicated nature of materials often necessitates a statistical approach to understanding and predicting their underlying physics. One such example is the empirical Weibull distribution used to describe the fracture statistics of brittle materials such as glass and ceramics. The Weibull distribution adopts the same mathematical form as proposed by Kohlrausch for stretched exponential relaxation. Although it was also originally proposed as a strictly empirical expression, stretched exponential decay has more recently been derived from the Phillips diffusion-trap model, which links the dimensionless stretching exponent to the topology of excitations in a glassy network. In this paper we propose an analogous explanation as a physical basis for the Weibull distribution, with an ensemble of flaws in the brittle material serving as a substitute for the traps in the Phillips model. One key difference between stretched exponential relaxation and Weibull fracture statistics is the effective dimensionality of the system. We argue that the stochastic description of the flaw space in the Weibull distribution results in a negative dimensionality, which explains the difference in magnitude of the dimensionless Weibull modulus compared to the stretching relaxation exponent.