Performance of Different Alloys by Assessing Epistemic and Aleatory Uncertainty in the Crack Growth Rates

Abstract

In probabilistic damage tolerance design, the choice of aluminum alloy may be influenced by the uncertainty/scatter in the crack growth rate and corresponding failure life. The epistemic uncertainty (due to modeling errors and limited test data) in the crack growth model may play a major role in the decision making along with the aleatory uncertainty (due to material variability). In this study, we estimate the epistemic uncertainty (via bootstrapping) and the aleatory uncertainty in the crack growth model parameters for 7475- T7351 and 7050-T7451 aluminum alloys. The choice between the alloys is made by comparing the reliability index estimated from respective crack growth failure life distributions. Finally, weight savings due to superior crack growth failure life of one alloy than other is estimated. Introduction The design of aircraft structures using probabilistic damage tolerance analysis (PDTA) involves modeling and propagation of uncertainties in the crack growth model inputs e.g. uncertainty in crack growth rate (i.e. da/dN vs. ΔK). We rely on experiments to estimate the best fit parameters of a crack growth model (e.g. Paris law, Walker equation). To estimate the uncertainty in fatigue crack growth life/failure life, these best fit parameters are treated as random variables and are the major source of aleatory uncertainty (due to material variability) and epistemic uncertainty (due to limited test data and model limitations). Selection of aluminum alloy for a particular damage tolerance application (e.g. wing spar) is typically based on the compromise between various material properties, such as fracture toughness, crack growth rate, yield strength, and stress corrosion cracking behavior. Of all these, the crack growth rate plays an important role in designing a light weight structure. A designer needs to choose an alloy that would satisfy a given probability of failure/risk constraint (e.g. 10 -4 ) with minimum weight penalty. The choice could possibly depend on the uncertainty (amount of scatter) present in the crack growth rate/model parameters and corresponding failure life. Therefore, it is important to quantify and compare the uncertainty (i.e. total uncertainty by combining aleatory and epistemic uncertainties) in the crack growth life for various aluminum alloys. In this study, we compare the uncertainty in crack growth life for the two most common aerospace aluminum alloys i.e. 7475- T7351 and 7050-T7451 plate materials. We compare both alloys under constant amplitude loading spectrum. The comparison is based on comparing the reliability indices (β’s) that are estimated from the crack growth failure life distribution for nominally identical specimen geometry. Then, weight savings due to better crack growth rate behavior is estimated i.e. an alloy with superior (larger) failure life will satisfy a given probability of failure constraint with lesser cross-sectional area, resulting in weight savings over other alloy. Test Data and Crack Growth Model Crack Growth Test Data The crack growth rate data (da/dN vs. ΔK) is derived from the crack length (a) vs. load cycle (N) data. The a vs. N data is obtained by laboratory testing of crack test specimens e.g. middle tension M(T) crack test specimen shown in Figure 1. A symmetric double through crack of total length (2c) is grown from a small hole of diameter (d) centered in a rectangular aluminum plate of width (W) and thickness (B). These tests are typically performed under 1 Doctoral Student, Dept. of Mechanical & Aerospace Engineering, kdsbhachu@gmail.com, AIAA Student Member. 2 Distinguished Professor, Dept. of Mechanical & Aerospace Engineering, haftka@ufl.edu, AIAA Fellow. 3 Associate Professor, Dept. of Mechanical & Aerospace Engineering, nkim@ufl.edu, AIAA associate Fellow. 4 Senior Engineering Specialist, Fatigue & Damage Tolerance Dept., churst@cessna.textron.com. 2 American Institute of Aeronautics and Astronautics constant amplitude loading (ΔP = Pmax – Pmin = constant) that are then used to predict the crack growth life under variable amplitude loading. In order to capture the stress ratio (R)/mean stress effects, tests are repeated at specific stress ratios (e.g. at R = 0.05, 0.3, 0.5, and 0.7). The stress ratio (R) is defined by the following relationship,