Stochastic multiscale models for fracture analysis of functionally graded materials

This article presents three multiscale models, including sequential, invasive, and concurrent models, for fracture analysis of a crack in a two-phase, functionally graded composite. The models involve stochastic description of the particle volume fractions, particle locations, and constituent material properties; a two-scale algorithm including microscale and macroscale analyses for determining crack-driving forces; and two stochastic methods for fracture reliability analysis. The particle volume fractions, defined by a generic inhomogeneous random field, are related to the intensity function of an inhomogeneous Poisson field, which describes the statistically inhomogeneous microstructure of a functionally graded composite. Two stochastic methods, the dimensional decomposition method and direct Monte Carlo simulation, have been employed for obtaining the probabilistic characteristics of crack-driving forces and reliability analysis. Numerical results indicate that the sequential and invasive multiscale models are the most computationally inexpensive models available, but they may not produce acceptable probabilistic characteristics of stress-intensity factors or accurate probability of fracture initiation. The concurrent multiscale model is sufficiently accurate, gives probabilistic solutions very close to those generated from the microscale model, and can reduce the computational effort of the latter model by more than a factor of two. In addition, the concurrent multiscale model predicts crack trajectory as accurately as the microscale model.