## 4 Results and Discussions
### Propagation of Elasto-Plastic Wave
This allows for us to make the following hypothesis, i.e., a shear band is formed every time mobile dislocations interact with barriers m times. With this hypothesis, we can estimate the average velocity of mobile dislocations at the onset of shear band propagation by substituting \(L=280,\ t_{s}=80\), and the onset time that is shown in Table 2 for \(t\) into the \(<v_{md}>\) expression (37). The fourth row of Table 2 lists the value of \(<v_{md}>\) estimated in this fashion for all puling rates. The dislocation density can be related to the average velocity of mobile dislocation, as follows (See Section 2.3 Dislocation Velocity on p. 17 of ref. [1] ). \[n=\frac{\gamma_{ex}}{b_{B}<v_{md}>} \tag{38}\]  
Here, \(n\) is the dislocation density \(\gamma_{ex}\) is externally imposed rate of deformation by the tensile machine and \(b_{B}\) is the magnitude of Burger's vector. Equation (38) indicates that the average velocity of mobile dislocations decreases in proportion to the mobile dislocation density. This explains the decrease in the shear-band propagation velocity with the passage of time for a given pulling rate. Using the above estimated \(<v_{md}>\), the average deformation rate (the tensile machine's pulling rate divided by the gauge length of the specimen, 25 mm ) and 0.286 nm for \(b_{B}\) (see p. 83 of ref. [47]), we can estimate the dislocation density at the beginning of the shear band propagation for each pulling rate. The results are shown in the fifth row of Figure 6.0 The dislocation density for pulling rates of 3 and 1 mm/min. is close to the initial (before deformation) dislocation density of copper discussed by [PERSON] et al. (See p. 25 of ref. [1]). When the pulling rates are lowered to 0.5, 0.3, and 0.1 mm/min., the onset dislocation density increases in this order by factor of two to five. These observations are consistent with the third row of Table 2, which indicates that the onset normal strain also increases in \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Pulling Rate (mm/min)** & **3** & **1** & **0.5** & **0.3** & **0.1** \\ \hline Onset time (s) & 6.6 & 31.1 & 150.1 & 456.2 & 2415 \\ Elongation (mm) & 0.55 & 0.51 & 1.25 & 2.28 & 4.02 \\ Normal strain (\%) & 2.2 & 2.0 & 5.0 & 9.1 & 16.1 \\ \hline Dislocation velocity (cm/s) & 1.92 & 7.16\(\times 10^{-1}\) & 1.77\(\times 10^{-1}\) & 6.03\(\times 10^{-2}\) & 1.16\(\times 10^{-2}\) \\ Dislocation density (cm\({}^{-2}\)) & 3.6\(\times 10^{4}\) & 3.3\(\times 10^{4}\) & 6.6\(\times 10^{4}\) & 1.2\(\times 10^{5}\) & 2.0\(\times 10^{5}\) \\ Solitary wave amplitude (mm/min) & 11.5 & 4.29 & 1.06 & 0.36 & 0.07 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Various quantities at onset of shear band propagation. Quantities in the top three rows are found directly from experimental data. Quantities in the bottom three rows are evaluated based on theories. this order. Figure 9 explicitly illustrates that the onset normal strain is approximately equal to the yield strain for pulling rates of 3 and 1 mm/min. and that the onset normal strain lags behind the yield point for pulling rate of 0.3 and 0.1 mm/min., where the amount of lag is greater for 0.1 mm/min. With the hypothesis that a solitary wave represents a shear band, we can discuss the parameters used in the solitary wave solution (29) semi-quantitatively. The quantity \(\sigma_{0}\) appearing in the solitary wave velocity expression (30) has been found in the neighborhood of 300 (See p. 179 of ref. [9]). Using the onset shear band velocity in Figure 8 and \(\sigma_{0}=300\), we can estimate the solitary wave amplitude \(a\) for each pulling rate, as follows. \[a=\frac{3c_{w}}{\sigma_{0}}=\frac{3c_{w}}{300} \tag{39}\]  
The sixth row of Table 2 shows the value of the solitary wave amplitude evaluated in this fashion in the unit of mm/min. It is seen that the solitary (velocity) wave's amplitude is of the same order as the pulling rate, which is understandable because the particle velocity is caused by the pulling action of the tensile machine.