Probabilistic Modeling of Brittle and Quasi-Brittle Fracture: First-Passage and Weakest-Link Analyses

Abstract

In this research, two new probabilistic models are proposed for strength distributions of brittle and quasi-brittle structures. The first model is a continuous probabilistic model based on first-passage analysis of random fields, which is referred to as the first-passage model. The model is first derived in a 1-dimensional setting and is applied to the strength statistics of poly-Si MEMS structures. Through the comparison with the experimental data, it is shown that the model is able to yield accurate predictions on strength distributions of MEMS structures of different sizes using the same model parameters. To improve the computational efficiency for predicting the strength distribution of MEMS devices, a renewal weakest-link model is developed. The model takes into account the detailed statistical information of the randomly distributed side-wall defects, which includes the random defect geometry, the random spacing between defects, and the local random material strength. The first-passage model is later generalized to higher dimensions for investigating the power-law behavior of strength distribution of brittle and quasi-brittle materials. It is shown that the power-law behavior of the left tail of structural strength distribution stems from the left power-law tail of material strength distribution, which is also mildly affected by the dimensionality of the analysis and the applied stress field. Flaw statistics (or the random stress field) introduces additional randomness to the structural strength, but does not dictate the power-law form of the tail distribution of structural strength. Lastly, the relationship between the internal length scale of the finite weakest-link model and the material length scales is investigated by analyzing the size effect on the mean structural strength. The mathematical form of this relationship is derived through the dimensional analysis, and the relationship is calibrated by matching the size effect curves yielded by the finite weakest-link model and the stochastic finite element simulations. It is shown that the internal length scale of the finite weakest-link model can be explicitly related to the Irwin characteristic length and the crack band width.