**1. Introduction**

Acoustic emission (AE) technology has the potential for on-line structural health monitoring; a desired procedure for evaluating material degradation in aircrafts. Acoustic emissions are stress waves that propagate through a material as a result of applied stresses. When a material is subjected to cyclic fatigue loading, AE signals may be generated frequently with cracking within the material. These waves can be detected by piezoelectric sensors when placed on the surface of the material. The characteristics of the AE signal are determined by the mechanism that generated the signal, and the means by which it travels through the material and the sensor that transforms the emission into the signal [1].

The most commonly used AE feature for fatigue is the AE counts, which is defined as the number of times that an AE signal amplitude exceeds a predefined subjective threshold value. Other common characteristics used to describe features of an AE hit include the signal's peak amplitude, risetime, and duration. The amplitude is defined as the maximum voltage value of the signal often reported in decibels, the risetime is the time between the first count and the count with peak amplitude, and the duration is the time between the first and last AE count. These features are illustrated in Figure 1, which shows a typical waveform from an AE hit with the associated features.

**Figure 1.** Characteristics of AE Signal.

Besides relating features from the AE signals to fatigue crack growth, some researchers supplement the AE signals with other crack growth indicators. For example, surface temperature mapping known as thermography has been used to enhance detection of crack initiation and growth. The driving factor for this development is that in addition to the release and propagation of stress waves that are detectable by AE sensors during fatigue crack growth, small amounts of thermal energy are also dissipated. Recent promising researches [2–5], built on the previous pioneering works on infrared t1hermography and AE signals, have also quantified fatigue crack growth. This paper, however, uses features of AE signals alone to describe fatigue crack growth behavior.

Mechanical structures typically operate under a wide range of loading scenarios. Previously proposed fatigue life models based on the AE signal properties reported by Talebzadeh and Roberts, [6]; Bianocolini [7]; and later by Rabiei & Modarres [8] have not been validated with respect to different loading conditions. These studies were also focused only on studying aluminum alloys. Different material and loading features, such as frequency and loading ratio, may also affect model predictions and can be verified through specially designed experiments.

Several attempts have been made to relate different AE parameters such as AE count, energy, and amplitude to fatigue crack growth, stress intensity factor range (Δ*K*), maximum stress intensity factor (*Kmax*) and crack growth rates [9–13]. These studies showed that the relationship between the stress intensity factor range and the number of AE counts can be captured by an equation with a form similar to the Paris-Erdogan equation [14–16]. A leading general model that relates the AE count rate to the crack growth rate was proposed by Bianocolini [7], and has the following form:

$$\frac{dc}{dN} = \kappa\_2 \left(\frac{da}{dN}\right)^{a\_1} \tag{1}$$

where, *c* is the AE count, *a* represents crack size, *N* is the load cycle, (*da*/*dN*) is the crack growth rate, (*dc*/*dN*) is the AE count rate, and *α*<sup>1</sup> and *α*<sup>2</sup> are the model parameters. Rabiei and Modarres [11] used a variation of Equation (1) in the form of the linear regression in Equation (2):

$$\log\left(\frac{da}{dN}\right) = a\_1 \log\left(\frac{dc}{dN}\right) + a\_2 + \varepsilon \tag{2}$$

where, the error term, *ε*, in Equation (2) accounts for the difference between the model prediction and the observed AE count rate. The model described by Equation (2) assumes that a small crack may be difficult to measure, and as the crack becomes larger, the measurement of crack length becomes more accurate [17]. In order to capture any changes in the error distribution, it was assumed that the error follows a normal probability density function with zero mean and a standard deviation, *s* (Equation (3)).

Based on a single experiment, Rabiei [17] concluded that *s* was independent of crack growth rate. To capture the dependence of the crack length on *s*, Rabiei [17] assumed that *s* follows a two-parameter exponential distribution that changes as a function of AE count rates. However, these assumptions

were not supported by the results of this research. The experimental results of this study suggested that *s* follows a normal distribution and is not a function of AE count rates; *s* ≈ *N*(*μs,σs*).

$$\varepsilon \approx \mathcal{N}(0, \mathbf{s}), \text{ s} \approx \mathcal{N}(\mu\_{\mathbf{s}}, \sigma\_{\mathbf{s}}) \tag{3}$$

The significance of the proposed model represented by Equations (2) and (3) is that once the model parameters are estimated using experimental data, this equation can be used to estimate crack growth rates of structures by monitoring AE signals and extracting the AE count rate from the observed signals.

Rabiei's model [17] was derived based on the results from one experiment with sinusoidal loading conditions of a maximum load of *Pmax* = 300 lbf (1334 kN), loading ratio of *R* = 0.1, and loading frequency of *f* = 10 Hz. The results have not been validated with respect to different loading conditions and were limited to one experiment on Al7075-T6. Therefore, the first step in this research was to study the consistency and variability of the model by performing several standard fatigue experiments using compact tension test specimens of the same aluminum alloy (Al7075-T6) subjected to cyclic loading with varied loading ratios and frequencies. This part of the research addressed model development and validation, with respect to changes in loading frequency and loading ratio, through statistical analysis of the crack growth data. In the second step, the material effect was studied based on three identical crack growth tests with Ti-6Al-4V titanium alloy (known as Ti-6-4). Experimental procedures and methods of analyzing the data were similar to those used for the Al7075-T6 samples.

This paper first discusses the details of the performed fatigue experiments, followed by the description of the data analysis methods, and finally presents results including model development and validation. In the final section, conclusions are summarized and suggestions are made for future work.
