Stochastic Extension of Nonlocal Macro–Mesoscale Consistent Damage Model for Fracture Behaviors of Concrete Materials

The nonlinearity and randomness in composite materials such as concrete present challenges regarding the safety analysis and reliability-based design of structures. Based on two-scale damage evolution and physically based geometry–energy conversion, the nonlocal macro–mesoscale consistent damage model (NMMD) shows a unique capability in dealing with the nonlinearity of crack evolution. In this paper, a stochastic extension of the NMMD model is proposed to analyze the stochastic fracture behaviors of concrete materials. The extended model uses the stochastic harmonic function (second kind) to represent the spatial variability in concrete properties and thus to investigate the influence of inhomogeneity in the cracking process. Numerical examples of three-point bending beams without defects and with initial cracks of various sizes demonstrate that the stochastic NMMD model is capable of not only capturing uncertain fluctuations in peak load but also simulating the random walk of the crack path with the instantaneous transition of fracture modes, as observed in experiments. In addition, the effectiveness of the stochastic NMMD model with only a single random field (i.e., Young’s modulus) also contradicts the conventional assertion that stochastic simulations of quasi-brittle fracture should contain at least two mechanical properties with spatial randomness. Finally, the investigation of fracture energy with stochastic fluctuations reveals that randomness resulting from heterogeneity can statistically improve the fracture toughness of concrete materials to some extent.