## 4 Results and Discussions
### Propagation of Elasto-Plastic Wave  
Figure 6 shows the propagation of \(u\) and \(v\) fringe patterns observed in post-yield stages of deformation in the same experiment, as in Figure 5. The three rows of \(u\) and \(v\) fringe patterns are formed successively with the same interval corresponding to an increment of 0.045% in the normal strain \(\epsilon_{xx}\). Each row represents a specific stage of deformation as indicated as \"stage 1\" through \"stage 3\" in the stress-strain curve at the top of Figure 6.  
Stage 1 (\(2.19\%<\epsilon_{xx}<2.41\%\))  
In stage 1, both \(u\) and \(v\) patterns travel upward at a constant rate. The \(u\) pattern is characterized by two half-circles located at the same horizontal position. The pair of half-circles occupies approximately a quarter of the vertical length of the specimen, and moves upward without changing their shapes. The \(v\) pattern is characterized by concentrated horizontal fringes located at approximately the same vertical position of the specimen as the pair of half-circles in the \(u\) pattern. The concentrated fringes move together with the pair of the half-circles in the \(u\) pattern. Figure 7 shows numerically obtained differential displacement patterns that result from the FEM computation conducted with the parameters listed in Table 1. The patterns labeled \(u\) correspond to the experimental \(u\) pattern discussed above, and the ones labeled \(v\) correspond to the experimental \(v\) patterns. The numerical results show qualitative agreement with the experimental patterns that are shown in Figure 6 in the following sense. (a) the \(u\) pattern consists of a pair of half-circles that move upward along the length of the specimen and (b) the \(v\) pattern consists of partially concentrated fringes that move in the same direction as the pair of half-circles in the \(u\) pattern. The FEM simply solves the wave equations for the horizontal (\(u\)) and vertical (\(v\)) components of the differential displacement without using a deformation model (except for the Poisson's effect). The agreement between the experimental and numerical results proves that the \(u\) and \(v\) fringe patterns observed in stage 1 represent the longitudinal and transverse waves. At \(t=t_{5}\), the numerical \(u\) pattern shows another  
Figure 5: (**a**) Stress–strain curve and (**b**) fringe images representing contour of differential displacement in (1), (2): Elastic stage (Equations (14) and (15)); (3)–(5): pre-yield stage and (6)–(8) post-yield stages (Equations (16) and (17)). Numbers that are placed above fringe images correspond to numbers in stress-strain curve. For each pair of fringe images left is \(u\)-fringes and right is \(v\)-fringes. pair of half-circles that are not seen in the experimental result. It is not clear why the second pair does not appear in the experiment. Stage 2 (2.86% \(<\epsilon_{xx}<\) 3.08%)  
In this stage, the \(u\) pattern still shows a pair of half-circles moving upward on the specimen. However, the half-circles are not horizontally symmetric, and the pair of half-circles are divided  
Figure 6: Change in wave motion. For each stage pairs of fringe patterns are arranged. Figure 7: Propagation of Finite Element Modeled (FEM) \(u\) and \(v\) wave-like pattern. by a group of approximately linear and slanted fringes. The slanted fringes indicate a shear band. The reason for this asymmetry is not fully understood at this time. However, it is interesting to note that the linear pattern possibly represents strong shear deformation and, within this region, the fracture condition (19) is satisfied in the recoverable mode (21). Perfectly-linear fringes make \(\omega\) defined by Equation (3) a constant, and therefore \(\
abla\times\mathbf{\omega}=0\). It is likely that the formation of the slant linear pattern (called the shear band) displaying the recoverable fracture condition is responsible for the sharp stress drop observed in the stress-strain curve in Figure 6. This type of sharp stress drop followed by immediate recovery is known as the serration.