# Dynamic Viscosity and Transverse Ultrasonic Attenuation of Engineering Materials
## 4 Discussion
However, the use of damping factors as the sole criterion for viscoelastic designation appears to require further verification regarding the applicability of the Knopoff equation to MHz frequencies. It is also necessary to find physical mechanisms of damping, especially in strong solids, such as Inconel 600, sapphire, and granite. These have damping factors (\(\eta\)) 2.5-, 5.4-, and 27-times higher than that of PMMA. While the dislocation bow-out mechanism [13] can possibly account for the case of Inconel 600, it is particularly difficult for sapphire with no grain boundaries and extremely high hardness (with a measured Vickers HN of 1760). Further studies are warranted. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Material Group** & **Ca Only** & **Ca + C\({}_{\textbf{a}}\)** & **Ca + C\({}_{\textbf{a}}\)** & **Ca + C\({}_{\textbf{a}}\)** & **Ca + C\({}_{\textbf{a}}\)** & **Ca + Low** & **Ca + High** & **Ca/C\({}_{\textbf{a}}\)** & **\(\geq\) 15** & **1.5 \(>\)**Ca/C\({}_{\textbf{a}}\)** & **\(\leq\) 0.67** & **0.67 \(>\)**Ca/C\({}_{\textbf{a}}\)** & **\(\eta\)** & **\(\eta\)** \\ \hline  & & **or C\({}_{\textbf{a}}\)** & **or C\({}_{\textbf{2}}\)** & & **d8/m/MHz** & **d8/m/MHz** & & & & & \\ \hline Ferrous & 22 (37) & 28 (35) & 2 (3) & 0 (6) & (0.1) & 164 & 17 & 21 & 15 & 6.69 \(\times 10^{-3}\) & 3.97 \(\times 10^{-3}\) \\ Non-ferrous & 41 (59) & 31 (29) & 7 (11) & 2 (0) & (0.4) & 341 & 54 & 16 & 11 & 6.52 \(\times 10^{-3}\) & 7.28 \(\times 10^{-3}\) \\ Polymers & 37 (48) & 0 (2) & 0 (0) & 0 (0) & 211 & 1830 & 37 & 0 & 0 & 3.27 \(\times 10^{-2}\) & 4.65 \(\times 10^{-2}\) \\ Fiber-reinforced plastics & 17 (25) & 0 (1) & 2 (0) & 0 (0) & 0 (0) & 1860 & 17 & 2 & 0 & 4.97 \(\times 10^{-2}\) & 6.52 \(\times 10^{-2}\) \\ Ceramics and rocks & 27 (30) & 38 (2) & 2 (2) & 0 (0) & (0.1) & 867 & 10 & 11 & 11 & 3.86 \(\times 10^{-2}\) & 1.15 \(\times 10^{-2}\) \\ Perronit of total tests & 65.3 (67.5) & 27.9 (21.6) & 5.9 (8.6) & 0.9 (2.3) & & 60.8 & 22.5 & 16.7 & \\ \hline \hline \end{tabular} Values in parentheses in columns 2.1 to 5 are for the longitudinal mode. Those in column 6 refer to the second lowest C\({}_{\textbf{a}}\). Tested sample counts for ceramics and rocks excluded fused silica since no attenuation was detected.  
\end{table} 
Table 8: General characteristics of attenuation parameters according to material groups. Atomic and molecular origins of low frequency damping in PMMA were listed in [61] and come from the molecular rotation of different parts of monomers (known as \(\alpha\), \(\beta\), \(\gamma\), and \(\delta\) relaxations), backbone chain bending, and sid \(\times\) chain tangling. Intermolecular vibration and density variation are other sources of energy loss [53; 54]. Molecular dynamics (MD) simulations on PMMA [62] successfully constructed an amorphous PMMA model and obtained the density, elastic modulus, and thermal expansion coefficient, but the damping part remains unreported. Additional works on this subject were collected [63], but none dealt with the damping simulation. Other MD works deduced stress-strain relations of PC, PE, and polystyrene and their methods can possibly determine the hysteresis behavior under lower cyclic stressing [64; 65]. [PERSON] et al. [64] emphasized important roles of shear stresses in driving structural changes in glassy polymers. This appears to contribute to substantially higher transverse attenuation found in all the polymer samples tested in this study. In addition to the MD studies, a different approach of shear transformation zone (STZ) theory [66; 67] can account for the shear deformation of amorphous silicon (obtained by a model calculation [68]). The STZ theory clearly showed a hysteresis effect at a near-yield stress. This behavior is likely to remain at low stresses. If so, the STZ theory in combination with the amorphous PMMA model [62] can be extended to common polymers, accounting for high \(C_{\text{dt}}/C_{\text{d}}\) ratios observed.