# Review on Structural Health Evaluation with Acoustic Emission
## 2 Signal Attenuation
### Complex Elastic Moduli Measurements  
[PERSON] et al. [50] developed an elegant ultrasonic method that is capable of determining the complex elastic moduli of PMMA discussed above. This method utilized iterative numerical inversion techniques and transmitted ultrasonic fields obtained for multiple incident angles on a plate sample. Either immersion or air-coupling technique can be used. This method yielded complex viscoelastic stiffness coefficients. The complex elastic moduli for PMMA were given in [PERSON] and Hosten [49], showing \(\eta_{1}\) = 0.023, \(\eta_{2}\) = 0.056, \(\eta_{66}\) = 0.026, and \(\eta_{12}\) = 0.046. These were measured at 0.3 MHz. The \(\eta\) data is tabulated in Table 1.  
For PMMA, many tests have been reported for \(\eta\) and for \(a_{\rm P}\). Figure 4 shows a plot of damping factor vs. frequency data from [51; 52; 53; 54; 55; 47; 56]. The value of \(\eta\) starts to rise at 0.001 Hz and the peak \(\eta\) of 0.09 is reached at 3 Hz, then decreasing at higher frequencies. These low frequency tests were in torsional mode (corresponding to \(\eta_{66}\) or \(\eta_{12}\)) and yielded 40 to 60% higher values than [PERSON]'s data [53], conducted in tension mode (equivalent to \(\eta_{1}\)), corresponding to \(a_{\rm t}\) and \(a_{\rm P}\) values, respectively, in terms of ultrasonic attenuation. Also plotted are \(\eta\) values converted from \(a_{\rm P}\) measurements at higher ultrasonic frequencies between 0.5 to 6.4 MHz [54; 55; 56]. The \(\eta\) values from ultrasonic attenuation are relatively unchanged at approximately 0.01 and match [PERSON]'s data near 1 MHz within 25%. In fact, [PERSON]'s early \(\eta\) value of 0.089 is valid from 0.29 to 30 MHz [57]. If the trend at lower frequencies continues to the MHz range, it can be expected that \(a_{\rm t}\) is 50% higher than \(a_{\rm P}\) in the low MHz region. When all these \(\eta\) values are compared among them, it is evident that the transmission field inversion method [49] produced at least a factor of two larger results. Independent evaluation of this method seems advisable as their other complex elastic moduli data have been utilized by several other research groups as will be discussed below. Note also that \(\eta\) values for PMMA decrease by a factor of two from 10 kHz to 30 MHz. Given ultrasonic attenuation coefficient \(a=\eta\tau/\lambda\), \(a\) values increase with frequency with their frequency dependence decreasing gradually. This behavior is expected in other engineering polymers. From the above survey and results, we can estimate the attenuation of guided waves when we have bulk wave attenuation data. Unfortunately, values of \(a_{\rm P}\) and \(a_{\rm t}\) are unavailable at typical AE frequencies under 1 MHz for most engineering materials because large samples are needed for attenuation measurements. Even in usual ultrasonic frequencies above 1 MHz, it is difficult to find even \(a_{\rm P}\) values for many materials. Still, high strength steels and Al alloys are qualitatively known to be good transmitters of ultrasounds. Thus, we can reasonably assume that AE signal attenuation follows the geometrical spreading and shows the \(1/\surd\)x behavior. This assumption cannot be used when excess damping conditions exist from inside or outside contact with liquids or other lossy matters [58; 59]. Then, we need to assess signal attenuation by traditional methods. Figure 3: Lamb wave attenuation curves with amplitude in dB and distance in mm. Data symbols are identical to those in Figure 2. (**a**) S\({}_{0}\)-mode propagation at 75 kHz on 6.4-mm thick PMMA. Red points are for the \(1/\surd\)x behavior plus \(a_{\rm L}\) of 25 dB/m; (**b**) mixed mode propagation at 380-kHz, fitting the \(1/\surd\)x behavior plus \(a_{\rm L}\) of 121 dB/m; (**c**) S\({}_{0}\)-mode propagation at 75 kHz on 4.6-mm thick PVC matched the \(1/\surd\)x behavior plus \(a_{\rm L}\) of 130 dB/m.