*3.1. Crack Growth Measurement*

The lengths of the pictured cracks were measured using an image processing software called ImageJ [24]. Crack measurement with the image processing software was calibrated using the scale ruler attached to the specimen. The accuracy of the ruler was 0.01 in. (0.254 mm); therefore, the scale error was estimated as ±0.005 in. (0.127 mm). One example of the process for measuring the crack length and matching the crack length with the cumulative AE counts is shown in Figure 4. By knowing the crack length and AE cumulative counts at numerous instances during the experiment, values for crack growth rate and AE count rate were estimated.

When the crack lengths were determined, the fatigue crack growth rates were approximated at different cycles using Equation (4):

$$\frac{da}{dN} = \lim\_{\Delta N \to 0} \frac{\Delta a}{\Delta N} \tag{4}$$

According to the observed data, the correlation between the AE count rates and stable crack growth rates follow the log-linear model of Equation (2). To properly compare the correlations estimated from all observed data obtained from different fatigue loading conditions on a similar scale, the correlation between the logarithm of crack growth rate and the normalized logarithm of *dc*/*dN* values was found. That is, the logarithm of every count rate was normalized to the range of the logarithms of minimum and maximum count rates observed in each test. With this normalization, the estimated parameters of Equation (2) were nearly identical for each test regardless of the loading

ratio or loading frequency and thus allowed all tests data to be combined to estimate one correlation model, as discussed in Section 3.3. Therefore, every *dc*/*dN* data point can be reversely expressed in terms of the corresponding normalized value and the two fixed minimum and maximum *log*(*dc*/*dN*) values. The correlation between normalized *log*(*dc*/*dN*) and *log*(*da*/*dN*) for different experiments at different loading ratios is shown in Figure 5a. Similarly, results for tests at different loading frequencies is shown in Figure 5b.

**Figure 4.** Example of crack length measurements paired with cumulative AE counts.

After the post processing of the AE data, the experimental results were used to determine the point estimate unknown parameters *α*<sup>1</sup> and *α*<sup>2</sup> in Equation (2) using regression analysis. The observed correlation of AE count rates versus crack growth rates and the regression analysis of the experimental data corresponding with different frequencies and different loading ratios showed that estimated parameters of the regression line are in the same range and do not have a considerable difference. It should be noted that the mean value of the regression parameters are calculated using all the experimental data and the estimation of the error term was implemented later in the probabilistic model development section. The point estimates of the linear regression model parameters for the different experiments are listed in Table 2. It should be noted that the fatigue lives (cycles-to-fracture) for Al7075-T6 samples were in the range of 3500 and 12,000 cycles. This range was determined to be 45,000 to 109,000 cycles for the Ti-6-4 samples.


**Table 2.** Regression parameters for individual test data, Al7075-T6.

**Figure 5.** Crack growth rate versus AE count rate for Al7075-T6 tests at (**a**) different loading ratios; (**b**) different loading frequencies—log scale. The base scale of *da*/*dN* is in/cycle.
