Inference of equivalent initial flaw size under multiple sources of uncertainty Shankar Sankararaman, You Ling, Chris Shantz, Sankaran Mahadevan * Department of Civil and Environmental Engineering, Vanderbilt University, Nashville, TN 37235, USA article info Article history: Received 1 April 2010 Received in revised form 16 June 2010 Accepted 30 June 2010 Available online 6 July 2010 Keywords: Fatigue life prediction Initial flaw size Crack growth Uncertainty quantification Bayesian inference abstract A probabilistic methodology is proposed in this paper to estimate the equivalent initial flaw size (EIFS) distribution accounting for various sources of variability, uncertainty and error, for mechanical compo- nents with complicated geometry and multi-axial variable amplitude loading conditions. A Bayesian approach is used to calibrate the distribution of EIFS, where the likelihood function is constructed from model-based fatigue crack growth analysis and inspection results. The variability, uncertainties and errors in the above procedures are quantified, and the distribution of EIFS is calibrated by explicitly accounting for the various sources of uncertainty. Three types of uncertainty are considered: (1) natural variability in loading and material properties; (2) data uncertainty, due to crack detection uncertainty, measurement errors, and sparse data; (3) modeling uncertainty and errors during crack growth analysis, numerical approximations, and finite element discretization. A Monte Carlo simulation-based approach is developed for uncertainty quantification in the crack growth analysis and for constructing the likelihood function of EIFS. The proposed methodology is illustrated by a numerical example. Ó 2010 Elsevier Ltd. All rights reserved. 1. Introduction Various sources of uncertainty, such as loading, material prop- erties and geometry of mechanical components, contribute to the non-deterministic behavior of fatigue crack growth. Fracture mechanics-based models, describing how the crack size increases under cyclic loading, are widely used to predict the fatigue life and evaluate the reliability of mechanical components. A prerequi- site of using the crack growth models is to determine the initial crack size. Two problems are encountered when the initial crack size is small. First it is difficult to measure the initial crack size if the limit of the inspection technique is reached. The second prob- lem is that the commonly used crack growth models are applicable only when the initial crack size is beyond micro-scale. In practice, empirically assumed initial crack in the long size region is used [1–3]. However, this leads to a very conservative fatigue life pre- diction if the actual initial crack size is small. The concept of equiv- alent initial flaw size (EIFS) was developed about 30 years ago to address the two problems above [4]. It is defined such that the pre- diction of a long crack growth model which starts from EIFS is in agreement with the actual fatigue crack growth [5]. Note that EIFS is not a physical quantity pertaining to the component, but a hypo- thetical flaw size that can be used as an initial crack size in long crack growth models. One approach to calculate EIFS is the back-extrapolation meth- od. Given some sets of experimental data (observed crack sizes at a certain time point), back-extrapolation from the given time instant to initial time instant is conducted based on fatigue crack growth models [5–7] or observed crack growth curves [8,9]. The crack sizes corresponding to the initial time instant are used to estimate the distribution of EIFS. The application of the back-extrapolation method is limited by its requirement for large amount of fatigue crack growth data. Insufficient test data will cause more uncer- tainty in the estimate of EIFS. A probabilistic methodology for the calculation of EIFS without back-extrapolation was proposed by Liu and Mahadevan [4] based on matching the infinite fatigue life predicted by the model with the experimental data from S–N tests. In this approach, some material properties (fatigue limit and threshold stress intensity factor) are treated as random variables. The distribution of EIFS is derived from the distribution of material properties, and hence crack growth analysis is avoided. Reasonable agreement was ob- served between this EIFS-based fatigue life prediction and the experimental results for specimens with simple geometry under uni-axial and multi-axial loading [4,10,11]. Components with com- plicated geometry under variable amplitude loading condition have not been considered yet in this line of research. Makeev et al. [12] and Cross et al. [13] applied statistical tech- niques, namely the maximum likelihood approach and the Bayesian approach, to calibrate the distribution of EIFS based on inspection data. In this approach, EIFS is treated as a model parameter that is calibrated through the observation of crack sizes at different num- bers of cycles. This approach is significantly different from earlier methods used for EIFS calculation using inspection data [5–9]; these are not based on statistics and use back-extrapolation of the 0142-1123/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2010.06.008 * Corresponding author. Tel.: +1 615 322 3040. E-mail address: sankaran.mahadevan@vanderbilt.edu (S. Mahadevan). International Journal of Fatigue 33 (2011) 75–89 Contents lists available at ScienceDirect International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue