## 5 Summary and Concluding Remarks  
Fringe patterns obtained from ESPI for a tensile experiment are analyzed in detail based on the field theoretical criteria of deformation stages. The wave equations that are derived by the field theory are solved numerically as a two-dimensional finite element model. The numerical results show semi-quantitative but clear agreement with the experimentally observed spatiotemporal behaviors of the differential displacement under elasto-plastic dynamics. This agreement confirms that the displacement field obeys the wave dynamics in the elasto-plastic regime, as predicted by the field theory. The analysis led to the following specific findings. 1. The fringe patterns obtained at approximately one-third of the yield strain exhibit the elastic deformation criterion. 2. The fringe patterns obtained at 80-90% of the yield strain indicate \(\
abla\times\mathbf{\omega}\
eq 0\), one of the conditions constituting the plastic deformation criterion. The \(u\) (the differential displacement component perpendicular to the tensile axis) fringe patterns obtained at approximately 95% of the yield strain show a circular pattern that represents the transverse-wave in the plastic stage. These observations indicate that plastic deformation starts 10-15% prior to the yield point. 3. When the normal strain reaches 2.19% past the yield strain of 1.0%, the \(u\) and \(v\) fringe patterns start to show traveling wave like behaviors. The numerical solutions to the wave equations confirm that these fringe patterns represent the transverse wave characteristics in the \(u\)-component and the longitudinal wave characteristics in the \(v\)-component of the differential displacement vector. The transverse and longitudinal waves propagate along the length of the specimen at the same speed. From this post-yield stage onward until the stress reaches the peak on the stress-strain curve, the wave motions repeat. 4. In the above-mentioned post-yield stage, the \(u\) and \(v\) fringe patterns are not uniformly distributed along the length of the specimen. This makes the \(u\) fringe consist of two half-circle patterns and the \(v\) fringe concentrated over approximately half the length of the specimen. The pair of half-circles in the \(u\) pattern is initially horizontally symmetric, and it propagates along the specimen length maintaining the half-circle shape. As the stress increases, the semi-circle patterns become asymmetric, and approximately linear slant fringe patterns (shear bands) appear between the asymmetrically located semi-circle patterns. The shear band propagates at the same speed as the semi-circle patterns. 5. The shear band propagation speed decreases towards the final fracture as a function of the elapsed time from the beginning of tensile loading. This behavior strongly indicates that the shear band formation and propagation are related to the mobile dislocation dynamics. The results of this study confirm the usefulness of the present field theoretical approach for diagnosis of deformation status. Especially, when combined with the ESPI for the visualization of  
Figure 11: Shear band pattern changes orientation. displacement field, it provides us with a quick way to make diagnosis. As a possible application of this combination of ESPI and wave dynamic diagnosis, we can consider the use of Artificial Neural Network (ANN) algorithm for an automated diagnostic system. We can use typical fringe patterns for the respective stages in one of the hidden layers of a pattern-recognition type of ANN algorithm [59] to train the system, so that it makes a judgment of a given ESPI fringe image for the stage of deformation. This is on our list of the future plan for this research. Conceptualization, S.Y. and T.S. ; methodology, S.Y. and T.S. ; software, S.Y. and C.M. ; validation, S.Y., C.M. and T.S. ; data curation, S.Y. and T.S. ; writing-original draft preparation, S.Y. ; writing-review and editing, C.M. ; funding acquisition, S.Y. All authors have read and agreed to the published version of the manuscript.