## 2 Dislocation damping
L is equal to [CD], and the radius of the curvature of the bowed-out segment (arc CBD) is denoted as R = [OB] = [OC] = [OD]. The angle 20 is defined by lines OC and OD. For this geometry, R is given by  
\[\text{R}=(\text{L/2})/\text{sin}\ \theta. \tag{9}\]The force on the bowed-out segment (arc CBD) is equal to \(\tau\) bL, and this is balanced by the opposing components of the line tension (\(=2\mathrm{a_{T}}\mathrm{\;Gb^{2}}\sin\theta\)), where \(\mathrm{a_{T}}\) is a line-tension constant of 0.5 and G the shear modulus. Expressing stress using a strain as with the shear modulus G, or \(\tau=\mathrm{a_{S}}\mathrm{\;G}\), and using Equation (9), we have [65]  
\[\tau\mathrm{bL}=\mathrm{a_{S}}\mathrm{\;GbL}=2\mathrm{a_{T}}\mathrm{\;Gb^{2}} \sin\theta=2\mathrm{a_{T}}\mathrm{\;Gb^{2}L/2R}, \tag{10}\]  
which reduces to  
\[\mathrm{a_{S}}\mathrm{\;R}=\mathrm{a_{T}}\mathrm{\;b}, \tag{11}\]  
Figure 4: Motion of pinned dislocation segments. (**a**) Dislocation shape at five positions on a loading cycle. (**b**) Applied stress vs. time. (**c**) Work done due to bowing out of dislocation segments vs. time. Figure 5: The geometry of a bowed-out dislocation segment. which relates the strain amplitude as to R; that is, given as and pinned segment length L, the values of \(\theta\) and R can be determined, allowing one to calculate the area of the bow-out \(\mathrm{A_{B}}\) (bounded by ACBD), as below. \[\mathrm{A_{B}}=\mathrm{R^{2}}(2\theta-\sin 2\theta)/2 \tag{12}\]  
This leads to the work done by each bow-out by multiplying \(\tau\) b/2 = as Gb/2. Here, a factor of half is needed because applied stress is cyclic, and another factor of half is also needed since the second quarter-cycle contributed no work. It should also be noted that \(\mathrm{A_{B}}\)/L corresponds to the average distance of dislocation motion. When the work done per half-cycle, as Gb \(\mathrm{A_{B}}\)/4, for each elemental volume of L\({}^{3}\) is doubled (for positive and negative motion) and divided by L\({}^{3}\), the unit volume value of work done per cycle is obtained. The applied strain energy per unit volume per cycle is G as\({}^{2}\)/2. The ratio of work done to the applied strain energy gives the damping factor \(\eta\) at the strain amplitude as by adding \(2\pi\) in the denominator, since \(\eta\) = (energy loss)/2\(\pi\) (stored energy). Then, one gets  
\[\eta=(\mathrm{a_{S}}\mathrm{G}\mathrm{B}\mathrm{A_{B}}/2)/(2\pi\mathrm{G} \mathrm{a_{S}}^{2}\mathrm{L}^{3}/2)=\mathrm{b}\mathrm{A_{B}}/2\pi\mathrm{a_{S} }\mathrm{L}^{3}. \tag{13}\]  
For \(\mathrm{a_{T}}\) = 0.5, G = 26 GPa, b = 0.3 nm, and L = 10 \(\mathrm{\SIUnitSymbolMicro\meter}\), the work done, applied strain energy, and damping factor were calculated for \(\mathrm{a_{S}}\) of \(2\times 10^{-8}\) to \(10^{-5}\) and plotted in Figure 6.  
For the strain amplitude range of \(2\times 10^{-8}\) to \(10^{-5}\), \(\eta\) values remain almost unchanged at 0.027. This is a very high value for metals and is high even for polymers. When the wave length of Al is used at 1 MHz, the \(\eta\) value corresponds to \(\alpha\) = 117 dB/m. The range of \(\mathrm{a_{S}}\) is wide and the maximum as corresponds to stress, which is one-third of the Orowan stress of unstable bow-out (or break-away stress). Here, the maximum bow-out distance was 0.86 \(\mathrm{\SIUnitSymbolMicro\meter}\) under 260 kPa, indicating the maximum dislocation velocity was 3.4 m/s at 1 MHz excitation. This is about 1000 times slower than the shear wave velocity of Al (3.1 km/s) and the dislocation motion can be regarded as quasi-static. On the other end of low as, the maximum bow-out under 0.52 kPa was 2.1 nm, with a dislocation velocity of 8.4 mm/s. Still, the same level of damping is predicted. Such dislocation velocities were well below the velocities limited by the dislocation damping coefficient, [PERSON] et al. [60] experimentally obtained B = 5.0 \(\times\) 10\({}^{-6}\) Pa-s at 300 K, while [PERSON] et al. [61] found B = 1.4 \(\times\) 10\({}^{-5}\) Pa-s at 342 K via a molecular dynamics (MD) approach.