## 4 Results and Discussions
### Propagation of Elasto-Plastic Wave
Propagation of the half-circle pair repeats a number of times between Stage 2 and Stage 3, where some of them exhibit the symmetric feature, like Stage 1, and others exhibit the asymmetry feature, like Stage 2. The symmetric and asymmetric features alternate in a random fashion. Although clear correlation has not been found between the serration and the formation of shear band at this time, this alternating behavior of the symmetric and asymmetric patterns involving the shear band formation makes us believe that the establishment of the recoverable fracture condition is closely related to the serration. The shear band (the slanted linear fringe pattern) can be interpreted as displaying the solitary wave solution (29). As mentioned above and confirmed in a number of experiments [14; 15; 16], the propagation speed of these solitary waves decreases with the development of deformation and the final fracture occurs when the wave becomes stationary. Thus, the slanted linear fringes can be used as an early indicator of deformation behavior in conjunction with the final fracture. The initial propagation speed is proportional to the pulling rate of the specimen. This proportionality has been qualitatively explained based on the field theory [18]. An interesting aspect that is related to this is that the temporal behavior of the decrease in the propagation speed is independent of the pulling rate and obeys a single function of time. Figure 8[49] shows the propagation speed of shear bands observed in the same type of experiment as above. The material was an aluminum alloy plate (A5083) [47]. The shear band propagation speed is plotted as a function of the elapsed time from the beginning of the tensile loading for five different pulling rates, 3, 1, 0.5, 0.3, and 0.1 mm/min. Table 2 lists the onset times (when the shear band appears for the first time since the beginning of the tensile loading), the total elongation computed as the product of the pulling rate and the onset time for each pulling rate, and the average normal strain evaluated by dividing the total elongation by the gauge length of 25 mm. Apparently, the shear band starts to appear at different total elongations (i.e., the averaged strain) among different pulling rates. On the other hand, as indicated by the trend line that is drawn in Figure 8, a single function fits the data for all of the five pulling rates. Numerical curve fitting indicates that a function in the form of \(A/(t+B)\) fits the best. Here, \(A\) and \(B\) are constants and \(t\) is the elapsed time. This particular form of  
Figure 8: (**a**) Shear band propagation speed as a function of time for different pulling rates. (**b**) Typical Electronic Speckle-Pattern Interferometry (ESPI) image showing a shear band. the function is of great interest, because the average speed of mobile dislocations, \(<v_{md}>\), can be expressed, as follows [1]. \[\textit{sinsinglesi}<v_{md}>=\frac{L}{(t+t_{s})} \tag{37}\]  
Here, \(L\) is the average distance between neighboring barriers and \(t_{s}\) is the interaction time with each barrier. The observed good fitting indicates that the shear band is formed in a process that is closely related with mobile dislocation's dynamics and its propagation speed is strongly related to the amount of the mobile dislocations. This argument is consistent with the prevailing theory regarding the micromechanism [50; 51; 52; 53; 54; 55; 56]. It is also consistent with the above-mentioned (see the paragraph under Equation (18)) self-driven nature of the evolution of deformation to fracture in association with an increase in the volume expansion rate \((\
abla\cdot\mathbf{v})\). The function \(A/(t+B)\) has the best fit to the experimental data in Figure 8 when \(A=280\) mm and \(B=8\) s. As the average distance between neighboring barriers for mobile dislocations, 280 mm is unrealistically long. However, notice that the value of this function remains unchanged if the same factor m is multiplied to \(L\), \(t\) and \(t_{s}\) as \(\text{m}L/(\text{m}t+\text{m}t_{s})\).