5 Discussion
In this study, theoretical solutions based on the wave function expansion method and Fourier transform were obtained for an inclusion in infinite space when subjected to a plane SH-wave. First, the steady-state response was analyzed using the wave function expansion method. Then, the Ricker wavelet was introduced as the transient disturbance. Finally, the Fourier transform was used to determine the distribution of transient dynamic stress concentration around the elliptical inclusion. The numerical results indicated that the dynamic stress concentration generated by the steady-state and transient incident waves was different. However, the dynamic stress concentration distribution in the two states was dependent on the elliptical axis ratio, incident wave number, difference in material properties between the medium and inclusion.
As reported in literature, the main content is analyzing angular stress concentration, while the radial stress concentration is often ignored. In this study, the angular and radial stress expressions under the steady-state incidence and transient incidence were obtained theoretically, and two types of distribution of dynamic stress concentration around the elliptical inclusion was calculated. It is found that regardless of the steady-state or transient incidence, for the inclusion stiffer than the medium, significant radial stress concentration appeared at both ends of the elliptical major axis. At the elliptical axis ratio of 10, the maximum steady-stateαξDSCF attained 3.081, and the maximum transient αξDSCFreached 7.869. The angular stress concentration was observed at both ends of the elliptical minor axis, but the value ofαηDSCF was slightly small. The maximum value of steady-stateαηDSCF was 0.764, and the maximum value of transientαηDSCF was 1.969. The closer the inclusion shape to crack, the larger the value ofαDSCF and the more significant the dynamic stress concentration. Therefore, the structural failure was likely to occur at both ends of the major axis and the minor axis of elliptical inclusion, and the radial stress concentration at both ends of the major axis of inclusion was more likely to cause structural failure. At this moment, the closer the inclusion shape to crack, the higher the possibility of the structural failure. However, for the inclusion softer than the medium, significant angular stress concentration was observed at both ends of the elliptical minor axis. The steady-stateαηDSCF achieved the maximum value of 2.079 at the elliptical axis ratio was 2.16, and the transient state αηDSCF attained the maximum value of 4.588 when elliptical axis ratio of 1.1. The radial stress concentration was noted at both ends of the elliptical major axis. When the axial ratio was 1.1, the maximum value of steady-stateαξDSCF attained 1.472, and the maximum value of transient stateαξDSCF reached 2.476. The more approximate the inclusion shape to circle, the larger the value ofαDSCF and the more significant the dynamic stress concentration. Therefore, the structural failure of inclusion was likely to occur at both ends of the major axis and the minor axis of elliptical inclusion, and the angular stress concentration at both ends of the minor axis of inclusion was more likely to result in the structural failure. At this time, the closer the shape of inclusion to circle, the greater the possibility of structural failure.
Compared with the wave function expansion method based on the Fourier-Bessel expansion and conformal transformation32, the Mathieu function was more convenient to deal with elliptical boundary problems, because it can avoid complicated boundary mapping, and the mathematical form was more concise. Moreover, the correctness of the theoretical derivation was verified by calculating distribution of dynamic stress concentration with the same material properties parameters between inclusion and medium. The results indicated that the spatial distribution of angular stress concentration was only related to the phase difference of the incident wave, and the maximum value of 1 was obtained in the vertical direction of the incident angle, which was consistent with the previous literature.10 The steady-stateαηDSCF gradually decreased with increasing the wave number for stiffer inclusion than medium. But when the inclusion was softer than the medium, the steady-stateαηDSCF oscillated with the change in the wave number, the greater the difference of material properties between the inclusion and medium, the more intense the oscillation. This phenomenon indicated thatαηDSCF had a high sensitivity to wave number, and the greater the difference in material properties between inclusion and medium, the higher the sensitivity.26
However, during the process of realizing the algorithm by mathematical software, the calculation of the Mathieu function was complex in some aspects. For example, under the limitation of boundary conditions, only the case of 0° incidence can be calculated. The wave number k of the incident wave was related to the q variable in the Mathieu function, the q value was related to the truncation series of the Mathieu function. In order to ensure accuracy, the truncation series must be changed when the k value was changed. In addition, the Mathieu function with no primitive function or the primitive function was difficult to express. As a result, only numerical integration methods can be used in Fourier integration, which greatly increased the amount of calculation. In order to ensure the accuracy of the results, this study compared the error between the sum of N and N+1 term. The Mathieu functions were generally truncated to the 8th or 9th to ensure accuracy. The Mathieu function was determined N=8-16, and the error between N=12 and N=16 was found to be less than 0.01%. Therefore, N=12 was selected to reduce unnecessary calculations while ensuring accuracy.