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.