3.3.2 Numerical results and analysis
The steady-state response of radial stress with different axial ratios and different wave numbers in the three cases was determined according to the case study. The numerical results were shown in Figs. 9-12. Figures 9-11 presented the distribution of steady-stateαξDSCF around the elliptical inclusion in three cases. As shown in Figs. 9-11, when ellipse approached circle and wave number was high, multiple extreme values ofαξDSCF appeared, and the distribution of αξDSCF on the front wave surface and the back wave surface was different. Although the distribution of αξDSCF was no longer as regular as Fig. (8), the distribution ofαξDSCF was still symmetrical about the x -axis. In Fig. 9,αξDSCF always had a maximum value at both ends of the elliptical major axis. As the angle was closer to both ends of the elliptical minor axis, the value ofαξDSCF became smaller, and a minimum value was observed at both ends of the elliptical minor axis. The maximum values of αξDSCF were 3.081, 2.616, 1.985, and 1.432, respectively, which were all larger than 1. This indicated that as the inclusion was stiffer the medium, the steady-state incidence caused significant radial stress concentration. As shown in Figs. 10 and 11, at the small radial coordinate, the distribution ofαξDSCF was similar to that in Fig. 9, but the maximum value was not larger than 0.28. At the large radial coordinate, the maximum and minimum values ofαξDSCF no longer occurred at both ends of the major and minor axis of ellipse, andαξDSCF on the front wave surface was more than that on the back wave surface. The maximum value of αξDSCF in Fig. 10 was 1.472, and the maximum value of αξDSCFin Fig. 11 was 1.016.
Figure 12 showed the steady-stateαξDSCF changed with radial coordinate and wave number at η =0° in three cases. Under the condition of constant wave number,αξDSCF of case 1 decreased slowly with the increase in the radial coordinate. This phenomenon in Fig. 12(a) was consistent with the changes ofαηDSCF , and the difference was that most of the value ofαξDSCF was larger than 1. In Figs. 12(c) and (e), under the condition of constant wave number, the variation trends of αηDSCF andαξDSCF in the radial coordinate were roughly identical. It implied that when the inclusion was softer than the medium, both αηDSCFand αξDSCF had high sensitivity to the radial coordinate. The more significant the difference between the material properties of the inclusion and medium, the greater their sensitivity.
As shown in Fig. 12(b), under the condition of constant elliptical axial ratio, the αξDSCF of case 1 decreased with increasing the wave number. But unlikeαηDSCF eventually approached 0,αξDSCF eventually approached 0.37. This revealed that for the inclusion stiffer than medium, the high wave number steady-state incidence additionally reduced the radial stress concentration. In Figs. 12(d) and (f), under the condition of constant elliptical axial ratio, the variation trends ofαξDSCF andαηDSCF in the wave number were roughly identical. This indicated thatαξDSCF also exhibited a high sensitivity to the wave number, and the larger the difference between the material properties of inclusion and medium, the greater their sensitivity.
Based on the foregoing analysis of the steady-state response, as the inclusion was stiffer than the medium, the dynamic stress concentration around the elliptical inclusion was dominated by the angular stress concentration at both ends of the minor axis.αDSCF gradually decreased with the increase in radial coordinate and wave number, but the final approach values ofαηDSCF andαξDSCF were different. However, when the inclusion was softer than medium, the dynamic stress concentration around the elliptical inclusion was dominated by the radial stress concentration at both ends of the major axis.αDSCF had high sensitivity to both the radial coordinate and the wave number. The greater the difference between the material properties of inclusion and medium, the larger the amplitude, the shorter the period, and the higher the sensitivity.