## 2 Dislocation damping
Another MD result [62] gives B = 2.8 \(\times\) 10\({}^{-5}\) Pa-s at 100 K (averaging screw and edge values). At 260 kPa, these B values predict dislocation velocities of 5.5 to 156 m/s, all exceeding the maximum dislocation velocities needed in the proposed bow-out mechanism of dislocation damping; that is, the bow-out damping mechanism operates in the quasi-static regime. When L is reduced to 1 \(\mathrm{\SIUnitSymbolMicro\meter}\), \(\eta\) values are almost unchanged at 0.027. Each elemental volume contributes less, but their number increase by 1000-fold. When \(\mathrm{a_{T}}\) is reduced, dislocation becomes flexible and damping increases substantially. Lower values of \(\mathrm{a_{T}}\) (of 0.1 to 0.5) are sometimes used in  
Figure 6: Applied strain energy per cycle (blue), work done per cycle (green), and damping factor (red circles) of dislocation bow-out vs. strain amplitude (= as). atomistic calculations [66]. At \(\rm{a_{T}}=0.1\), the work done is 90% of the applied elastic energy, increasing \(\rm{\eta}\) by five-fold; that is, \(\rm{\eta}\) varies in proportion to \(\rm{a_{T}}\). It appears that low line tension of \(\rm{a_{T}}\) below 0.1 is unrealistic. Thus, as long as dislocations are mobile with low stress, oscillating dislocations provide the basis for rationalizing the linear frequency-dependent damping behavior. However, the lack of effect of pinning length reduction on \(\rm{\eta}\) values is surprising, as this was initially thought to offer a clue in understanding cold working effects. The explanation of two contradictory cold working effects will have to be sought elsewhere [67]. The dislocation bow-out mechanism proposed above is based on shear deformation, so the attenuation is for the shear wave propagation. However, the attenuation also applies to the longitudinal wave mode, as the normal and shear stresses (or strains) are related via the Taylor factor, M (or 1/M) [68]. Consequently, the strain energy and external work terms are unaffected by the coordinate transformation involved; that is, the attenuation coefficients from dislocation damping are identical in the longitudinal and shear modes of wave propagation. [PERSON] [8] and [PERSON] [41] made the only known direct comparison of attenuation coefficients between the longitudinal and shear modes, reporting 2.5- to 8-fold increase for the shear attenuation for five metallic materials. However, most of these were taken at 10-15 MHz, where the Rayleigh scattering caused the attenuation. Further study is needed to find if similar differences persist at lower frequencies, as was the case of 2017 Al alloy [8]. If so, the above prediction of the bow-out mechanism will have to be reconsidered. Many mechanisms can lower attenuation to the ranges of observed \(\rm{\alpha}\) values, including higher lattice resistance to dislocation, dislocation-dislocation interactions, solute atmospheres, and distribution of the second phase. However, the dislocation bow-out mechanism introduced here does provide a new approach to account for the linear frequency dependence of the attenuation coefficient in non-viscoelastic solids. The main limitation of this mechanism is its inability to explain the attenuation of hard solids. These have low dislocation mobility due to their intrinsic strength and their linear attenuation behavior remains unresolved.