*2.1. Guided Wave Attenuation*

In using AE to examine structures, we typically deal with two-dimensional wave propagation since thick-walled structures—like nuclear pressure vessels and concrete dams—are exceptional cases. For theoretical developments and early experimental results, see Viktorov's book from 1967 [20] or numerous other books that followed dealing with wave propagation, e.g., Rose [21]. In most instances, AE signals propagate along the surface as Rayleigh waves or through thin-walled structures as Lamb waves. These waves are collectively known as guided waves. When they spread on two-dimensional structures, signal loss occurs from geometrical spreading (with inverse square root dependence on travel distance, x, or 1/√x dependence). Additionally, attenuation is caused from material absorption and scattering with exponential decay, or exp(−*a*x) with attenuation coefficient *a*. Combined, signal amplitude decreases following (1/√x)·exp(−*a*x). For dispersive Lamb waves, signal loss also arises from dispersion or frequency-dependent wave speed, which spreads vibration energy over a longer period, reducing signal amplitude [13,20]. When the surfaces are covered with liquids or other damping matters, attenuation occurs from vibration energy leakage. These appear as additional *a* values. Mal et al. [22] gave an extension of wave propagation theory to anisotropic plates with dissipation given in terms of quality factors, *Q*. They presented several examples of attenuation in fiber reinforced composite plates.

Press and Healy [23] in 1957 gave theory and experimental confirmation for Rayleigh wave attenuation. Measured Rayleigh attenuation coefficients (*a*R) for PMMA were nearly linear with frequency and was 55 dB/m at 100 kHz. Viktorov [20] provided a parametric equation for *a*<sup>R</sup> and listed three more *a*<sup>R</sup> values for aluminum (Dural or 2017 alloy), glass and polystyrene (15.4, 73.7, 422 dB/m at 1 MHz, respectively). In 1964, Zhukov et al. [24] derived theoretical expressions for Lamb wave attenuation coefficients, *a*L. These guided wave attenuation coefficients, *a*<sup>R</sup> and *a*L, depend on the attenuation coefficients of longitudinal and transverse bulk waves, or *a*<sup>p</sup> and *a*t. Zhukov et al. [24] also measured *a*<sup>L</sup> values for the 0th and 1st symmetric and asymmetric modes, or S0, S1, A0, and A1 modes on low carbon steel plates (C = 0.15%). The measured *a*<sup>L</sup> values ranged from 3.8 to 5.7 dB/m at 1.3 MHz and were comparable to their longitudinal and shear wave attenuation coefficients, *a*p = 3.7 and *<sup>a</sup>*<sup>t</sup> = 3.9 dB/m. At zero thickness limit for S0, *<sup>a</sup>*<sup>L</sup> <sup>=</sup> <sup>√</sup><sup>2</sup> *<sup>a</sup>*t. Pressure-vessel and pipeline steels are known for their low bulk wave attenuation of *a*<sup>p</sup> = 1 to 10 dB/m at 2 MHz as tabulated in Krautkramer's book [25], which are reduced further at lower frequencies. Such low attenuation values for Al and steels below 10 dB/m were reported by Mason and McSkimin [26], Roderick and Truell [27], Kamigaki [28], and Papadakis [29,30] among others at frequencies up to 20 MHz. These data were obtained using directly bonded, low-loss quartz transducers. Some of these tests also included *a*<sup>t</sup> measurements. Thus, guided waves are expected to propagate with low loss when materials possess low *a*p values.

Wave attenuation is characterized using several different parameters. In the ultrasonic and AE fields, attenuation coefficient *a* is commonly used to represent an exponential decay. We use two units for *a*; One is dB/m and the other Np/m with 8.686 dB = 1 Np. Np stands for Nepers, a non-dimensional unit, and is useful in numerical computation. This *a* is also related to damping (or loss) factor η by η = *a*λ/π, where λ is the wave length. Here, η is the ratio of energy dissipated per cycle to maximum energy stored per cycle, and is also equal to loss tangent, tan δ. This tan δ is defined as the ratio of imaginary part (E") to real part (E') of a complex modulus, E\* = E' − *i*E" with *i* <sup>2</sup> <sup>=</sup> −1. The damping factor (or loss tangent) is often used in dealing with vibration damping at lower frequencies. Another parameter is quality factor *Q*, defined as the inverse of η (or tan δ). See e.g., Kinsler et al. [31] and Cai et al. [32].

#### *2.2. Attenuation Measurements on Large Metallic Structures*

Graham and Alers [33] reported in 1975 the first quantitative study of AE signal attenuation on pressure vessels, showing that signal amplitude (expressed in dB scale) decreased linearly with distance of propagation, except near the signal source where the slope of the decrease was steeper. The frequency they examined was 100 to 850 kHz and the wall thickness of the pressure vessels was 100–130 mm. Thus, the signals examined were Rayleigh waves. Selected data was replotted against distance (in the logarithmic scale) as shown in Figure 1. Of the five plots, the data at 100 and 200 kHz followed the inverse square root distance dependence, indicating absorption loss was negligible. For the two higher frequency cases at 400 and 600 kHz, additional attenuation term of 1.7 or 2.1 dB/m accounted for the observed deviation from the 1/√x dependence. Data at 850 kHz was inadequate, but it seems to show even higher attenuation. For a given plate thickness, a limiting frequency exists, below which Rayleigh waves do not exist. For the 100-mm thick steel, it is 82 kHz, as shown in the Appendix A.

Pollock [34] reported AE signal attenuation for 30-kHz signals on a pipeline of nearly 300-m length. A replot of this data (with blue + symbols) in Figure 2a exhibits the same 1/√x behavior (indicated by a black line) with small additional attenuation of 0.17 dB/m. Data points for the 1/√x plus 0.17-dB/m attenuation are shown by red circles. Here, Lamb waves propagated on a relatively thin pipe wall (of a few cm thickness). Blackburn [35] reported attenuation data for large gas cylinders. These cylinders were 12-m length, 0.6-m diameter, and 14.3-mm wall thickness and made of heat-treated 4130 steel (quenched and tempered after fabrication). His data for a 3AAX tube at 150 kHz is plotted in Figure 2b. Most data points fit the 1/√x behavior except a few points deviated lower, suggesting possible effects of signal absorption at large distances. In these two studies, dispersion loss was minimal.

More attenuation measurements have been published recently. The data of Baran et al. [36] for 30-kHz signals on a pipeline up to 100-m length are shown in Figure 2c. Their results are similar to the Pollock case with a slightly higher attenuation of 0.34 dB/m. Sofer et al. [37] tested 150-kHz signals from pencil-lead breaks on steel sheet and plate, getting only the 1/√x behavior since the maximum distance was 2 m. Their cylindrical block did exhibit attenuation of 17 dB/m in addition to the 1/√<sup>x</sup> behavior. Thus, these observations fit with the theory and early experiments [20,24].

**Figure 1.** Attenuation data (amplitude in dB vs. distance in m) on steel pressure vessels at five frequencies from Graham-Alers [33]. From top to bottom: 100 kHz (blue), 200 kHz (red), 400 kHz (purple), 600 kHz (orange), and 850 kHz (green). Signal source was a white noise generator. Lines drawn represent 1/√x dependence.

Two other works were outside the above framework. CETIM group [38] examined large penstocks in a hydropower plant covering the length of 40 m. Their low and medium frequency attenuation studies resulted in the inverse-distance behavior. This may be due to many thick flanges for these particular penstocks that appear different from the common design of long pipe sections. El-Shaib [39] reported Lamb wave attenuation on a steel plate, but his signal energy values decreased with 1/√x, implying negative amplitude attenuation. It is likely that his energy data was already converted to amplitude values.

**Figure 2.** Attenuation data (amplitude in dB vs. distance in m) on steel structures. (**a**) Signal attenuation on a pipeline at 30 kHz by Pollock [34]; (**b**) Signal attenuation on a gas cylinder at 150 kHz by Blackburn [35]; (**c**) Signal attenuation on a pipeline at 30 kHz by Baran et al. [36]. Lines drawn represent 1/√x dependence. Measured data points are shown by + (blue) and modeled points are in filled circles (red) in (**a**,**c**).

#### *2.3. Laboratory Attenuation Measurements*

As a part of guided-wave sensor studies at UCLA [40,41], Lamb wave attenuation was measured for large aluminum (Al) plates (6.4-mm thick 1100 Al and 12.7-mm thick 6061 Al), steel (3.2-mm thick 410 stainless), soda-lime glass (4.7-mm thickness), PMMA (6.4-mm thickness), and polyvinyl chloride (PVC, 4.6-mm thickness). For the three metal and glass plates, the inverse-square root distance behavior prevailed with a few exceptions (when combined modes started to split at larger travel distances at some frequencies). The maximum travel distance was 500 mm and the frequency range was from 100 to 1500 kHz. The result of the 1/√x behavior for 410 stainless steel (SS) was surprising because this steel (along with pure Fe and pure Ni) was expected to show high bulk wave attenuation due to its magnetic properties [42]. However, Papadakis [43] reported only moderate attenuation for a comparable 416 stainless steel (*a*p = 20 dB/m at 4 MHz). Papadakis [43] did find Ni to have a high *a*p of 120 dB/m at 2 MHz. Drinkwater [44] calculated Lamb wave attenuation curves for glass, showing less than 0.1 dB/m attenuation below 0.5 MHz for a plate (3.9-mm thickness). Thus, the 1/√x behavior found here for glass confirms the calculation. In highly attenuating polymeric plates, signal levels were strongly reduced as shown in Figure 3. At 75 kHz on 6.4-mm thick PMMA (Figure 3a), S0-mode waves propagated with symmetric excitation and showed the 1/√x behavior plus *<sup>a</sup>*<sup>L</sup> of 25 dB/m. At 380 kHz on PMMA (Figure 3b), seven Lamb modes are expected with the group velocity of 0.8 to 1.1 mm/μs according to dispersion curve calculation. Since the duration of excited signals was about 50 μs, mode separation was not observed and the entire wave packets were analyzed. Attenuation *a*<sup>L</sup> beyond the geometrical spreading was higher at 121 dB/m. A 4.6-mm thick PVC plate exhibited even higher attenuation, reflecting its higher bulk wave attenuation [45]. As shown in Figure 3c, 130-dB/m attenuation was observed beyond the geometrical spreading at 75 kHz. For PMMA, the values of *a*<sup>p</sup> and *a*t below 100 kHz were available from resonant ultrasonic spectroscopy [46,47]. Assuming that the damping factor (η) values of 0.035 and 0.025 reported for 50 and 60 kHz, respectively, hold at 75 kHz, *a*<sup>p</sup> and *a*<sup>t</sup> are found as 27.3 and 52.0 dB/m for PMMA at 75 kHz. Using the Zhukov theory for *a*<sup>L</sup> and calculated coefficients given, the attenuation of S0-mode waves was obtained as 33.7 dB/m for the frequency-thickness product of 0.48 MHz-mm for PMMA thickness of 6.4 mm. The observed *a*<sup>L</sup> value for S0-mode is 25 dB/m, so *a*<sup>L</sup> matching is good between theory and experiment. Castaings and Hosten [48,49] obtained complex elastic moduli for PMMA and predicted Lamb wave attenuation for five lowest modes. At 75 kHz, *a*<sup>L</sup> for S0 was 21.5 dB/m, while *a*<sup>L</sup> values ranged from 145 to 300 dB/m at 380 kHz. Thus, theory and experiment agree reasonably well also. When the observed *a*<sup>p</sup> and *a*<sup>t</sup> values are inserted to the Viktorov equation [20] for Rayleigh wave attenuation, *a*<sup>R</sup> = A *a*<sup>p</sup> + (1 − A) *a*<sup>t</sup> = 51.0 dB/m as A = 0.05 with Poisson's ratio of 0.37 for PMMA. This is in good agreement with 47 dB/m at 75 kHz, obtained by Press and Healy [23].

**Figure 3.** *Cont.*

**Figure 3.** Lamb wave attenuation curves with amplitude in dB and distance in mm. Data symbols are identical to those in Figure 2. (**a**) S0-mode propagation at 75 kHz on 6.4-mm thick PMMA. Red points are for the 1/√x behavior plus *<sup>a</sup>*<sup>L</sup> of 25 dB/m; (**b**) mixed mode propagation at 380-kHz, fitting the 1/√x behavior plus *<sup>a</sup>*<sup>L</sup> of 121 dB/m; (**c**) S0-mode propagation at 75 kHz on 4.6-mm thick PVC matched the 1/√x behavior plus *<sup>a</sup>*<sup>L</sup> of 130 dB/m.

#### *2.4. Complex Elastic Moduli Measurements*

Castaings 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 Castaings and Hosten [49], showing η<sup>1</sup> = 0.023, η<sup>2</sup> = 0.056, η<sup>66</sup> = 0.026, and η<sup>12</sup> = 0.046. These were measured at 0.3 MHz. The η data is tabulated in Table 1.

For PMMA, many tests have been reported for η and for *a*p. Figure 4 shows a plot of damping factor vs. frequency data from [47,51–56]. The value of η starts to rise at 0.001 Hz and the peak η of 0.09 is reached at 3 Hz, then decreasing at higher frequencies. These low frequency tests were in torsional mode (corresponding to η<sup>66</sup> or η12) and yielded 40 to 60% higher values than Thakur's data [53], conducted in tension mode (equivalent to η1), corresponding to *a*<sup>t</sup> and *a*<sup>p</sup> values, respectively, in terms of ultrasonic attenuation. Also plotted are η values converted from *a*<sup>p</sup> measurements at higher ultrasonic frequencies between 0.5 to 6.4 MHz [54–56]. The η values from ultrasonic attenuation are relatively unchanged at approximately 0.01 and match Thakur's data near 1 MHz within 25%. In fact, Hartman's early η 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*<sup>t</sup> is 50% higher than *a*<sup>p</sup> in the low MHz region. When all these η 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 η values for PMMA decrease by a factor of two from 10 kHz to 30 MHz. Given ultrasonic attenuation coefficient *a* = ηπ/λ, *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*<sup>p</sup> and *a*<sup>t</sup> 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*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/√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.


**Table**

**Figure 4.** Damping factor of PMMA over 11 decades of frequency, peaking at 3 Hz, from the literature. Torsional damping: blue X [47], blue circle [51], green circle[52];longitudinaldamping:green+[53],redcircle[54],purplesquare[55],purplediamond[56].

#### *2.5. Survey of Ultrasonic Attenuation of Metallic Alloys*

Table 2 lists ultrasonic attenuation of engineering alloys, noting material parameters given in the original articles. This list attempted to cover all the attenuation data available for engineering alloys, but some data were omitted for obvious errors in methods used. Large variations are sometimes seen for similar alloys, but detailed reevaluation of measurement methods is needed to explore their causes [67]. For example, Klinman et al. [68] obtained *a*p of 60 dB/m at 5 MHz for an annealed 0.15%C steel with 20 μm grain size, but later Klinman et al. [69] reported for a similar sample *a*<sup>p</sup> of 670 dB/m at 10 MHz. Papers from Harwell [70,71] also reported 500 to 1350 dB/m attenuation for annealed low C steels at 8–10 MHz. Such large differences in *a*p at 5 and 10 MHz may arise from the frequency dependence of *a*p. However, these appear to be strange, since Papadakis [29,72] obtained *a*p = 12 dB/m for annealed 4150 steel and below 10 dB/m for a tempered martensitic bearing steel (52100 steel) over 1 to 10 MHz. Serious evaluation of these diverging results has not been conducted so far, but Papadakis' data are more reliable as he used directly bonded quartz disc transducers, while later studies used damped ultrasonic transducers and water immersion [68–71]. Magnetic effects could be a factor, but are less than 10 dB/m at 1 MHz and not large enough [73,74]. Most of the references dealt with the longitudinal wave attenuation. Recently, Hirao, Ogi, and Ohtani [75–79] have measured shear wave attenuation using non-contact electromagnetic sensors. For example, they found *a*t = 116 dB/m at 5 MHz for 0.15%C steel with 49 μm grain size, almost doubling Klinman's *a*<sup>p</sup> data at 5 MHz cited above [68]. Their *a*<sup>t</sup> results in combination with longitudinal attenuation data allow one to estimate guided wave attenuation using the theories discussed in Viktorov [20]. The number of engineering alloys examined for *a*t, however, is still limited.

In contrast to polymers, in which hysteretic effects due to molecular rearrangements are dominant [18], ultrasonic attenuation of crystalline metals and ceramics mainly comes from absorption, Rayleigh scattering, and stochastic scattering [26,80]. Absorption effect is similar to polymers with linear frequency dependence, though mechanisms are different. Both of the scattering contributions depend on frequency with a power law and the exponents are 4 and 2. Rayleigh scattering varies most with grain size. In Table 2, attenuation measurements that used low-loss quartz discs directly bonded to samples are marked with Q while non-contact electromagnetic measurements are marked with E. The rest utilized immersion, buffer rod, or direct contact methods. The unmarked group needs assumptions regarding signal loss at sample interfaces, where errors may be generated. Generazio [81] examined ultrasonic reflection coefficients and showed large variations and dependence on interface thickness and contact pressure. In view of low attenuation found in guided wave propagation dominated by the 1/√x dependence, it is likely that most structural steels used in pressure vessels, tanks, and pipelines have low values of *a*<sup>p</sup> and *a*t. Thus, high bulk wave attenuation of ultrasonic waves must be reexamined since high attenuation reports have mostly originated from immersion test procedures that included a plane-wave assumption in the analysis. This last point also needs further study because the sound fields ahead of a commonly used piston transducer suffer from divergence and diffraction [31,72].



*Appl. Sci.* **2018**, *8*, 958



measurement method using quartz discs and E using electromagnetic transducer. 3 GS: grain size, GSc: pearlite colony size or prior austenite grain size, VHN: Vickers hardness number, RHC:Rockwellhardness-C-scale,M:martensite.

#### *2.6. Guided Wave Attenuation on Fiber-Reinforced Composites*

Lamb wave experiments were also conducted at UCLA using fiber reinforced plastics (FRP). Figure 5a,b show two cases for an FRP with woven glass-fiber rovings and epoxy matrix (2.2 mm thickness, density 1.9 g/cm3, 0.38 fiber fraction). Attenuation data for S0 mode along the fiber direction at 60 and 100 kHz are plotted against (log) distance. The observed data can be attributed entirely to the 1/√x dependence at 60 kHz, while 100-kHz data matches the 1/√x dependence plus 10.8 dB/m attenuation. These attenuation coefficients are similar to *a*<sup>p</sup> of 4.4 dB/m at 100 kHz for a unidirectional GFRP along the fiber (converted from η of 0.007) [90]. The above findings implies glass-fiber reinforced plastics (GFRP) behave similarly to common structural metallic alloys below 100 kHz. The same attenuation data can also be described conventionally with an attenuation coefficient of 21.2 dB/m (60 kHz) or 33.1 dB/m (100 kHz) including the geometrical spreading as shown in Figure 5c. These conventional attenuation coefficients are lower than *a*p of 135 dB/m for another GFRP (with random mat) at 100 kHz, with its *a*p increasing to 400 dB/m at 2 MHz [91]. However, this data was in the direction normal to fibers and not directly comparable. When attenuation due to absorption is high, it is convenient to include signal spreading in attenuation parameters in NDE applications.

**Figure 5.** Lamb wave attenuation data for S0 mode along the fiber direction of an FRP. (**a**) Data at 60 kHz match with 1/√x dependence; (**b**) data (blue +) at 100 kHz fit to the 1/√x dependence plus 10.8 dB/m attenuation, shown by red circles; (**c**) the same attenuation data (in dB) plotted against linear distance. Blue points: 60 kHz with the slope of 21.2 dB/m. Red points: 100 kHz with 33.1 dB/m. This FRP was used in an AE study [92].

Castaings and Hosten [48] used complex elastic moduli from [49] and predicted Lamb wave attenuation for unidirectional GFRP, giving S0-mode *a*<sup>L</sup> value at 100 kHz of 17 dB/m. For this case, Castaings and Hosten [49] reported η<sup>11</sup> = 0.047, η<sup>22</sup> = 0.043, η<sup>33</sup> = 0.033, η<sup>55</sup> = 0.05, η<sup>66</sup> = 0.038, η<sup>12</sup> = 0.033, and η<sup>13</sup> = 0.036 (fiber direction is the 3-direction and fiber volume 0.6). Neau et al. [61]

reported another set of ηij for a GFRP, measured by the Castaings method. This set showed the values of η<sup>33</sup> and η<sup>66</sup> to be 70 and 240% higher. These η values by the Castaings method (listed in Table 1) are higher than other published values of η, determined from conventional vibration damping methods at lower frequencies. Crane [93] thoroughly reviewed test methods and results. He also gave additional results on damping of composite materials. See also [60], which listed η values from the 1980s as in [93]. For unidirectional GFRP with 0.6-fiber volume fraction, longitudinal damping factor η in the fiber direction was 0.004–0.006, about six-times smaller than the ultrasonic data above [49,61]. Crane [93] also noted that epoxy matrix has η of 0.022. Vantomme [94] obtained even lower damping factor values, e.g., η<sup>1</sup> = 0.002 (or 0.0013 from [60]) and noted η for a single glass fiber to be 0.0015. Data in [60] gave η<sup>2</sup> of 0.008 and η<sup>12</sup> of 0.011 for E-glass/DX210 epoxy. These GFRP data are mostly lower than the epoxy data in Table 1 [60]. These damping data were all taken under 1 kHz and increasing trends with frequency were reported. If the main contribution to damping comes from the matrix, as existing theories postulated [91], the increase with frequency is likely to be insignificant considering the decreasing trend of damping factor noted on PMMA [42]. In fact, the data of *a*<sup>p</sup> = 400 dB/m at 2 MHz cited earlier [91] correspond to the damping factor of 0.02. Thus, the Castaings–Hosten determination of GFRP damping factors that averaged to 0.043 apparently suffers from overestimation of a factor of two or more.

Attenuation and damping studies for carbon-fiber reinforced plastics (CFRP) have been conducted since the 1970s. In his exhaustive review, Crane [93] showed that longitudinal damping factor in the fiber direction (η1) was 0.001–0.005 for unidirectional CFRP with 0.6-fiber volume fraction. Transverse damping factor normal to the fiber direction (η2) was approximately 0.01. CFRP data from [60] were similar to GFRP values given above and match with the Crane values. A newer study confirmed these results [95]. These damping studies were made at low frequencies below 20 kHz using flexural bending of long beam samples. A recent work also verified η<sup>1</sup> of ~0.001 using a longitudinal resonance technique with pultruded 0◦ samples [96]. At 2 MHz, η<sup>1</sup> or η<sup>2</sup> of 0.01 corresponds to ultrasonic attenuation coefficient (*a*p) of 50 or 182 dB/m in the direction parallel or normal to fibers. Using quartz disc transducers, Kim [97] evaluated *a*<sup>p</sup> and *a*<sup>t</sup> of UD-CFRP (XA-S/1138) including those along the fiber direction over 1.8 to 9 MHz. At a fiber fraction of 0.6 and 2 MHz, *α*<sup>p</sup> was 55 and 450 dB/m, parallel and normal to fibers, while *α*<sup>t</sup> exceeded 1050 dB/m. Biwa et al. [98] made theoretical and experimental studies of attenuation for UD-CFRP (TR30/340) with epoxy matrix varying fiber volume fractions up to 0.6. At 2 MHz, *a*p was 430 dB/m normal to fibers, about 10% higher than their theory. In both studies [97,98], the values of *a*t were much higher than those of *a*p. When the observed *a*<sup>p</sup> of 450 dB/m is converted to η2, we get η<sup>2</sup> of 0.025, about twice the low frequency data. An earlier work by Williams et al. [99] obtained η<sup>1</sup> and η<sup>2</sup> values of 0.014 and 0.049 along and normal to fibers at 2 MHz for a CFRP (AS/3501-6). They used multiple samples of square cross section (9.5 × 9.5 or 12.7 × 12.7 mm2) and 3.8- to 127-mm length to get their <sup>η</sup><sup>1</sup> and <sup>η</sup><sup>2</sup> values. Even ignoring a diffraction correction [100,101], *α*<sup>p</sup> along fibers at 2 MHz is 70 dB/m and is only 27% higher than Kim [97]. For *a*<sup>p</sup> normal to fibers, it is 892 dB/m without diffraction correction and is about twice those of Kim [97] and Biwa et al. [98]. We also conducted an ultrasonic attenuation measurement at 2.25 MHz using an immersion method and obtained α<sup>p</sup> normal to fibers of 704 dB/m for CFRP (G50/F584) in a cross-ply layup, corresponding to η<sup>2</sup> of 0.034. Along the fiber direction, the same CFRP yielded η<sup>1</sup> of 0.02 at 0.5 MHz. This CFRP sample was from our earlier AE study [102]. The differences in attenuation among CFRP are partly due to fibers and resins used, but may also be from methods used. Thus, we expect η<sup>1</sup> = 0.01~0.02 and η<sup>2</sup> = 0.02~0.05 for UD-CFRP at low-MHz ultrasonic frequencies.

Using a torsion pendulum method, Adams [103] evaluated the damping η for single carbon fibers. For polyacrylonitrile (PAN)-based fibers, η was 0.0013, while pitch-based fibers (after stretching) had η of 0.0028. Ishikawa et al. [104] examined η for single carbon fibers of 13 different combinations of tensile strength and elastic modulus. These were newer fibers with improved properties and both PAN- and pitch-based fibers were tested for torsional damping. For the low η group, the values of η were about 0.025 for low strength fibers (under 2 GPa tensile strength). The middle η group had

η = 0.035 ± 0.003 and the tensile strength varied from 2.5 to 6 GPa, including both high strength PAN fibers and high modulus meso-phase pitch fibers. The last group, or the high η group, showed high damping with η = 0.05 to 0.08 and possessed high strength of 3.8–6.3 GPa. This high damping in the newer high-performance carbon fibers is surprising since these η values are three to four times larger than epoxy matrix and nearing those of some elastomers. Earlier damping results from the 1980s were on CFRP with fibers of low to medium strength in today's classification. Thus, these fibers had values of η under 0.04 and in combination with epoxy (η = 0.022), resultant η<sup>2</sup> is consistent with the observed η<sup>2</sup> of 0.02~0.05. Another check can be made by measuring ultrasonic attenuation. If the high damping values of 0.05 to 0.08 persist at low MHz frequencies, we should expect α<sup>p</sup> of 900–1500 dB/m at 2 MHz (2300–3600 dB/m at 5 MHz) of ultrasonic attenuation. This level of α<sup>p</sup> was reported on CFRP with T700 fiber. Olivier et al. [105] measured α<sup>p</sup> of 3300 dB/m at 5 MHz for 8-ply unidirectional CFRP, but with 1.7% porosity. Our recent preliminary measurement on a CFRP with T700 fiber (2–10 mm thickness, made from Toray 2501 prepregs; fiber fraction of 0.4) showed α<sup>p</sup> = 72 or 600 dB/m at 0.45 MHz in the fiber direction or normal to fiber direction. These α<sup>p</sup> values correspond to η<sup>1</sup> and η<sup>2</sup> of 0.05 and 0.13, respectively, and indeed imply that newer high-performance fibers have higher intrinsic damping in comparison to fibers made before the mid-1990s. The high damping behavior is beneficial for many engineering applications, but poses a challenge for SHM. Ishikawa et al. [104] related high torsional damping to amorphous carbon phase. Yet there seems to be no clear reason why newer high-performance carbon fibers possess high acoustic damping. Note that amorphous fused silica has very low damping. These ultrasonic studies should be repeated including the evaluation of neat resin and further research is needed for the origin of damping.

Complex elastic moduli of CFRP plates have been measured with the Castaings method. Four sets of ηij values are tabulated in Table 1. Note that fibers are oriented along the 3-axis for Neau et al. [61], but along the 1-axis for Matt [62] and Calomfirescu [63,64]. In three cases for CFRP-UD samples, damping along the fiber axis averaged to 0.069 while shear damping normal to fibers 0.098. These damping values are again two or more times higher than η<sup>2</sup> values of comparable CFRP-UD samples. The complex elastic moduli were then used with higher order plate theory to calculate attenuation coefficients for S0 and A0 modes. Calomfirescu [63] obtained for a UD plate along the fibers (0◦) α<sup>L</sup> of 27 and 150 dB/m at 300 kHz (45 and 250 dB/m at 500 kHz) for S0 and A0 modes, respectively. Along the fiber normal (90◦) direction, corresponding values were 75 and 108 dB/m at 300 kHz. These attenuation values are beyond geometrical spreading. A few other calculations resulted in much higher attenuation values [16,61,62] than Calomfirescu results.

The S0 value along 0◦ of Calomfirescu [63] was much higher than our measurements on AS4/3506-1 CFRP plates [106]. In our CFRP study, three types of lay-ups (unidirectional, cross-ply, and quasi-isotropic) were used and complex attenuation behavior was found when the wave propagation direction shifted from the fiber orientations. For the UD plate, S0-mode attenuation value along the fiber direction (0◦) at 300 kHz was 4.8 dB/m, which was slightly less than that of Al plate (5.8 dB/m). Since geometrical spreading was not separated and receiver size was 8 mm, the same test was repeated using a KRN sensor of 1 mm size. Figure 6a gives attenuation data for S0 and A0 modes at 100 kHz, which are represented by attenuation coefficients of 20 dB/m for S0 and 38 dB/m for A0 mode. When the same data is plotted in a log-log graph, only geometrical spreading effect or 1/√x behavior is found without additional attenuation. No signal loss was observed for S0-mode at 300 or 500 kHz, as shown in Figure 6b. This behavior appears to come from a sharp directivity for the UD plate observed previously [106]. In contrast, attenuation for A0-mode at 300 or 500 kHz was substantial. Values of *a*<sup>L</sup> were 78 and 178 dB/m beyond geometrical spreading for A0-mode propagation along the fiber direction, as shown in Figure 7a,b. Thus, the *a*<sup>L</sup> values for A0 are within a factor of two with the Calomfirescu calculation [63]. However, *a*<sup>L</sup> for S0 at 0◦ vanished in our tests beyond geometrical spreading.

**Figure 6.** (**a**) UD-CFRP attenuation for S0 (blue) and A0 (red) modes at 100 kHz, amplitude in dB scale; (**b**) UD-CFRP attenuation for S0 mode at 300 kHz (blue) and at 500 kHz (red), amplitude was calculated using the square root of the wave packet energy. Information on the UD-CFRP plate was given in [106].

**Figure 7.** UD-CFRP attenuation for A0 modes vs. (log) distance. Observed data (blue +), modeled attenuation of geometrical spread plus attenuation (red circle), geometrical spread (gray circle). Amplitude in dB scale. (**a**) Attenuation of 78 dB/m for 300 kHz; (**b**) attenuation of 178 dB/m for 500 kHz. Amplitude was calculated as in Figure 6.

Schmidt et al. [65] used another approach in characterizing CFRP properties. They measured Lamb wave propagation utilizing air-coupled ultrasonic techniques and deduced the dispersion curves and attenuation behavior. The data was then used to construct an analytic model. Their modeling relied on higher order plate theory and the attenuation utilized hysteretic model where the damping coefficients (ηij) are independent of frequency. The values of ηij were given in Schmidt et al. [66] and listed in Table 1. It appears that an UD plate is marked 45◦ considering Cij values. Assuming this to be the case and noting that their ηij stand for C"ij, some of these are two-to-five times smaller than the corresponding values from the Castaings method [49]. However, off-fiber parameters of η22, η33, and η<sup>23</sup> are grossly overestimated with η<sup>23</sup> = 0.26 being five-times higher than η<sup>12</sup> of PMMA. In dealing with different composite layups used, lamination theory was incorporated. An example given shows calculated attenuation coefficients for S0 and A0 modes on a quasi-isotropic CFRP plate. At 122 kHz, they obtained attenuation coefficients for S0 and A0 modes to be 4.3 and 50.2 dB/m beyond geometrical spreading. Another set of attenuation coefficients for S0 and A0 modes were reported graphically (CFRP layup was not given, but it seems to be UD). At 300 kHz, these were 31 and 140 dB/m, respectively. These *α*<sup>L</sup> matched with their experimental values. These were higher than our data that included geometrical spreading effects [106], and our new test data discussed above showing no attenuation at 100 kHz. Only the A0 data at 300 kHz matched within a factor of two. Schmidt et al. [65] included geometrical spreading in their modeling. This needs to be probed, however, since properly modeled calculations should predict 1/√x behavior from the viscoelastic parameters for quasi-isotropic CFRP, and a directional behavior for unidirectional CFRP. More comparative studies are obviously needed to obtain representative CFRP wave propagation characteristics.

## *2.7. Summaries*

Section 2 considered signal attenuation, which often limits the use of AE inspection in many SHM applications. Since waves in most structures move as guided waves, Section 2.1 collected available theories for attenuation in isotropic media and described a general behavior, citing early experiments. Section 2.2 reviewed published wave propagation experiments, showing that available results can be rationalized well using the theories from Section 2.1. Section 2.3 reported Lamb wave propagation experiments conducted in laboratory scale for confirmation using elastic and viscoelastic plates. Results matched theoretical predictions. Section 2.4 discussed a new approach for attenuation studies using complex elastic moduli, which were obtained by the Castaings method. All the available results were collected in Table 1. Results for PMMA were then compared with damping factor determination, which had been accumulated over many years. It was found that the Castaings method prediction appears to overestimate the damping factor by a factor of two, suggesting the need for independent verification. Section 2.5 covered the ultrasonic attenuation of metals. For metallic alloys, structural attenuation behavior can be predicted when their attenuation coefficients, both longitudinal and transverse, are known. However, the attenuation data is limited and all the accessible values were tabulated in Table 2. More studies are needed, especially for commonly used structural alloys and for transverse attenuation coefficients that require special instrumentation. Section 2.6 dealt with the attenuation behavior of fiber composites. For these anisotropic media, only limited data sets are available and over short propagation distances. Calculations relied on higher order plate theory and complex elastic moduli from the Castaings method. This approach appears to be valid based on comparison with Lamb wave attenuation data and represents substantial advances in wave analysis. However, some of the damping factor data are apparently two or more times higher when compared to the corresponding values from ultrasonic attenuation measurements. Again, further studies are required to clarify the attenuation behavior of highly variable, anisotropic composite structures. Note that most available complex elastic moduli data sets lack manufacturing data on tested composite plates, making comparison difficult. Detailed material identification must accompany sophisticated mechanical characterization.

The above discussion demonstrated that modeling of guided waves for CFRP plates has advanced substantially. However, more refinements are needed in calculation procedures and validation of damping coefficients. While Rayleigh and hysteretic damping models produced realistic attenuation results that matched experiment [16,65], Kelvin–Voigt model as used in, e.g., [65], should be discarded for its physically unrealistic assumption. Additionally, the Kelvin–Voigt model introduces an arbitrary parameter (characteristic frequency) in damping calculation. Critical evaluation of the Castaings and Schmidt methods [49,66] for damping parameters is highly desirable to achieve a unified predictive

method. As noted above, the Castaings method [49] gives damping factors twice higher than other methods and some ηij were also excessive in Schmidt et al. [66].

These basic studies point to difficulties in practical testing of FRP structures except at lower frequencies. As in the Sandia report [11], Weihnacht et al. [107] discussed testing of large FRP components. In one of their tests, they needed to place 64 sensors over a 38-m long blade at 3 to 5 m sensor spacing. Such a sensor count demonstrates tough challenges facing real time composite inspection. Thus, the strategy introduced by CARP [4] of using low (60 kHz) and high (150 kHz) frequency sensors in tandem remains valid for global and local AE source location.
