A phase-field model of fracture with frictionless contact and random fracture properties: Application to thin-film fracture and soil desiccation

We present a new derivation for a phase-field model of cohesive fracture that allows for fully-damaged surfaces to properly transmit tractions under frictionless contact conditions. The model is derived from an energy minimization standpoint, and the governing equations are presented in a general Allen–Cahn form, unifying both brittle and cohesive fracture models. A novel elastic energy split is proposed to enforce frictionless contact conditions along the regularized crack set. A fixed-point iterative algorithm is used to solve the system of equations, and an active-set strategy is incorporated to enforce the crack set irreversibility. The model is then applied to study the mechanical-fracture coupling in the fracture of thin films and soil desiccation problems in which intricate crack networks form. We present results for one-dimensional, two-dimensional, and fully three-dimensional configurations, and examine the relationship between the generalized driving energies for fracture and the resulting fracture patterns. We also investigate the impact of stochastic spatial variations in material parameters on fragment statistics and fracture network morphology. To this end, we develop a random field model for fracture properties and address a forward propagation problem. The versatility offered by the probabilistic framework is finally highlighted by considering a statistical inverse problem based on physical experiments. It is shown, in particular, that the formulation allows key characteristics of observed crack patterns to be reproduced with reasonable accuracy.