Brittle fracture reliability by probabilistic finite elements

The fusion of the probabilistic finite element method (PFEM) and reliability analysis for probabilistic fracture mechanics (PFM) is presented. The PFEM is extended to PFM using an enriched element that has the near crack-tip singular strain field embedded. Static condensation is used to solve for modes I and II stress intensity factors, and the adjoint approach to PFEM is employed for evaluating the derivatives of the stress intensity factors with respect to the random variables. Statistical moments (e.g., expectation, covariance, and correlation) of stress intensity factors are calculated for uncertainties in load, material properties including fracture toughness, component geometry, and crack geometry (i.e., crack length, orientation, and position). In addition, the first-order probability of brittle fracture is calculated. In order to calculate the probability of fracture, an optimization procedure is employed to determine the reliability index. The methodology is demonstrated on two mode I fracture examples. The fusion of PFEM and reliability for fracture mechanics is computationally quite efficient and provides a powerful tool for the design engineer.