*3.4. Model Validation*

In order to validate the developed AE model, model predictions of crack growth rate were compared against the validation experimental data set. For a given value of AE count rate, a prediction of crack growth rate was estimated based on the developed AE-based model. Available information captured from Al7075-T6 experimental data was used as the input to the Bayesian estimation procedure, as the model was developed based on Al7075-T6 data. The Bayesian estimation approach updated the model prediction with the experimental results. With this Bayesian estimation inference, uncertainties in the experimental values were propagated in the model and resulted in model prediction uncertainty assessment. The prediction results were then compared against the true crack growth rates obtained by validation experiments.

In the proposed model validation methodology [31], both model prediction and experimental results are considered to be estimations and representations of the true values, given some error as it is shown in Equations (8) and (9):

$$\frac{X\_i}{X\_{\varepsilon,i}} = F\_{\varepsilon,i};\ F\_{\varepsilon} \sim LN(b\_{\varepsilon\prime}s\_{\varepsilon}) \tag{8}$$

and

$$\frac{X\_i}{X\_{m,i}} = F\_{m,i};\ F\_m \sim LN(b\_{m,i}s\_m) \tag{9}$$

where *Xi* is the true value, *Xe,i* and *Xm,i* indicate the experimental results and model prediction, respectively. *Fe* is the multiplicative error of the experiment with respect to true value, and *Fm* is the multiplicative error of the model prediction with respect to true value. A multiplicative error of the experiment with respect to the model prediction is defined by Equation (10):

$$\frac{X\_{\mathbf{c},i}}{X\_{m,i}} = \frac{F\_{m,i}}{F\_{\mathbf{c},i}} = F\_{\mathbf{t},i} \tag{10}$$

Since both *Fm,i* and *Fe,i* distributions are lognormal, the distribution of *Ft,i* would also be lognormal with mean and standard deviation of (*bm* − *be*) and *<sup>s</sup>*<sup>2</sup> *<sup>m</sup>* + *s*<sup>2</sup> *<sup>e</sup>*, respectively. In this approach, the likelihood used is shown in Equation (11):

$$L(X\_{\varepsilon,i} \mid X\_{m,i}, b\_{\varepsilon}, s\_{\varepsilon} | b\_{m,i} s\_{\mathfrak{m}}) = \prod\_{i=1}^{n} \frac{1}{\sqrt{2\pi t \left(\frac{X\_{\varepsilon,i}}{X\_{\mathfrak{m},i}}\right) \sqrt{s\_{\mathfrak{m}}^2 + s\_{\varepsilon}^2}}} \exp\left(-\frac{1}{2} \times \frac{\left[\ln\left(\frac{X\_{\varepsilon,i}}{X\_{\mathfrak{m},i}}\right) - (b\_{\mathfrak{m}} - b\_{\varepsilon})\right]^2}{s\_{\mathfrak{m}}^2 + s\_{\varepsilon}^2}\right) \tag{11}$$

Using the validation sets of data, the validation approach was implemented and the results are discussed in this section. For more information about the validation method, refer to the paper by Ontiveros [31]. For simplicity, the distributions of model predictions were reduced to a mean value and compared one-to-one with the experimental results. The mean and standard deviation of *Fe*, which are *be* and *se*, were determined from the unbiased experimental error of ±1%. The values determined were −0.00002 for *be* and 0.002 for *se*. The summary statistics for the marginal posterior *pdf* of parameters *bm* and *sm* as well as the distribution of *Fm* are presented in Table 4.



The model uncertainty bounds for the crack length estimation can be determined from the 2.5 and 97.5 percentile of the multiplicative error of *Fm*. The resulting upper bound on reality was calculated as 14%, while the lower bound is −10%. These results are presented graphically in Figure 10. It can be noticed that the mean values of the model prediction are standing on the upper bounds of the experimental error, and the value of *Fm* for the model prediction is greater than 1. The value of *Fm* around 1.01 suggests a very small bias in the AE model to under predict the true crack growth rate. Therefore, the estimation of the true crack growth rate given by the model prediction is expected to be slightly higher. The results show that to correct the model, the predictions must be increased by a factor of 1.01.

**Figure 10.** Comparison of AE model prediction and experimental results (log *da*/*dN*). WB indicates Bayesian regression analysis results using WinBugs.

Using the experimental data points for *log*(*da*/*dN*) and assuming an experimental error of ±1%, the 45-degree solid line shows the difference between the model prediction and the "true" *log*(*da*/*dN*) (with the 45-degree line showing the perfect match). However, because of the experimental data scatter and a slight bias expressed by *Fm* (showing systematic model error) between the true *log*(*da*/*dN*) and regression models of Figure 9, the true *log*(*da*/*dN*), with a 95% confidence level, will be somewhere as high as 14% above the model prediction and as low as 10% below this prediction.

As expected, the validation results showed good agreement between model predictions and experimental observations. The validated AE model can be used in real time monitoring of large cracks that subsequently improves the structural health management.

#### **4. Conclusions**

This research focused on the AE model for large crack growth assessment. In order to establish the AE signal feature versus the fatigue crack growth model and study the consistency and accuracy of the model, several standard fatigue experiments were performed. A set of tests were performed using standard Al7075-T6 test specimens subjected to cyclic loading with different amplitude and frequencies. A previously proposed relationship between the crack growth rate and AE signal features generated during crack growth was modified and validated. The AE-based crack growth model was found to be independent of the loading condition and loading frequency. These findings validate the previous work by Rabiei [17] as the proposed model is independent of the loading ratio and frequency.

Based on three identical tests with Ti-6-4, it was concluded that while the model parameters are material-dependent, the linear model form depicting the relationship between the crack growth rate and AE count rate remains valid when the material is changed.

The obtained experimental data was uncertain in nature due to considerable uncertainties in the optical crack detection method and measurement errors associated with the utilized crack sizing technique. The procedure of probabilistic model development and validation were discussed, and uncertainties of the model were investigated. To deal with uncertainties, a Bayesian approach was used to consider systematic and random errors in the model by capturing the effect of uncertainties. This approach provided a framework for updating the distribution of the model parameters.

Development of the proposed AE monitoring technique reported in this paper facilitates for the prognostics and life predictions of the structure. The developed methodology can be utilized for continuous in-service monitoring of structures and has proven to be promising for use in life predictions and assessment for structures subject to fatigue loading. Ultimately, these predictions can be used to define the appropriate inspection policies and maintenance schedules.

To update the model for any material variation, quantitative material properties that correlate to the AE count rate and crack growth rate could be identified. This process would be extensive due to the numerous material properties that affect a material's failure mechanisms including fracture toughness, modulus of elasticity, and yield strength. Once properties are identified, numerous types of materials could be tested to validate and update the model to account for material variations. In addition, the model is based on tests using the same specimen geometry of a specific plate thickness. Since crack growth behavior can also be dependent on the thickness of the material, the AE response and subsequently the model would be dependent on material thickness and specimen geometry. Future research may also focus on experimental studies to account for the effects of material thickness.

**Author Contributions:** A.K. developed and performed the Al7075-T6 acoustic emission and fatigue crack growth experiments while C.S. performed the Ti-6-4 experiments. A.K. and C.S. each performed the initial data analysis steps including crack growth measurements, de-noising, and AE count rate calculations for each of their respective experiments. A.K. performed the Bayesian analysis and model parameter development for all tests. A.K. wrote the majority of the paper, while C.S. contributed to several sections and assisted in editing. Finally, M.M. was the academic research advisor in this work and provided guidance throughout the research and edited the paper.

**Conflicts of Interest:** The authors declare no conflict of interest.
