Fracture of beams with random field properties: Fractal and Hurst effects

The classical problem of peeling a beam off a substrate is studied through a re-examination of Griffith's fracture criterion in the presence of multiscale random properties. Four types of wide-sense homogeneous Gaussian random fields of the vector {Young's modulus E, surface energy density γ}, parametrized by the beam axis, are considered: Ornstein–Uhlenbeck, Matérn, Cauchy, and Dagum. The latter two are multiscale and allow decoupling of the fractal dimension and Hurst effects. Also calculated is the variance of the crack driving force G with any given type of random field in terms of the covariances of E and γ, under either fixed-grip or dead-load conditions. This investigation is complemented by a study of the stochastic crack stability which involves a stochastic competition between potential and surface energies. Overall, we find that, for Cauchy and Dagum models, the introduction of fractal-and-Hurst effects strongly influences the fracture mechanics results. Notably, while the Cauchy and Dagum models represent a more realistic scenario of random fields, given the same covariance on input, the response on output is strongest for Matérn, then Ornstein–Uhlenbeck, then Cauchy and, finally, Dagum model.