Statistics of fracture for an elastic notched composite lamina containing weibull fibers—Part II. Probability models of crack growth

Analytical probability models and useful approximations are developed to study the effects of statistical variation in fiber strength on the statistics of the fracture process, fracture resistance, and overall strength distribution for an elastic composite lamina with a transverse notch of N contiguous broken fibers (0 ≤ N ≤ 51). In Part I, we generated representative fracture strength distributions with the aid of Monte-Carlo simulation, coupling realistic micromechanical stress transfer and Weibull fiber strength. We also examined various statistical aspects of composite failure, and showed that variability in fiber strength can manifest in a nonlinear failure mechanism in an otherwise elastically deforming composite. Inaccurate estimates obtained from using both a unique fiber strength fracture criteria and a weakest fiber model pointed to the need for statistical approaches that account for crack plane microcracking and stable crack growth. Mechanisms responsible for flaw tolerance in the short notch regime and for toughness in the long notch regime, were identified as being important in modeling. Based on the probabilities of simple sequences of failure events, we develop here probability models for crack growth, including one for microcracking and one for fracture resistance, which capture the various statistical features of failure modes and composite fracture strength. We employ these methods to determine practical upper and lower bounds on composite failure probability, to model R-curve behavior observed for low fiber Weibull modulus values γ and large N, and to predict the unnotched composite strength distribution. In particular we find that fracture toughness grows slowly with N, scaling as (ln N)17. Analytical approximations allow for fracture strength calculations and estimates of the amount of crack growth prior to failure at all probability levels, especially in the region of high reliability. Both the models and analytical approximations compare very favorably with the Monte-Carlo results. Distinct differences in the statistics of the fracture process emerge when γ is reduced from the range γ ≥ 5 to γ = 3. Analytical calculations suggest that an important transition in behavior occurs at γ = 2, where damage at the crack tip becomes much more distributed.