3.2 Steady-state response of angular stress
When computing the steady-state response of angular stress with different axial ratios and different wave numbers in three cases according to the case study, the numerical results were shown in Figs. 4-7. Figures 4-6 presented the distribution of the steady-stateαηDSCF around the elliptical inclusion in the three cases. As shown in Figs. 4-6,αηDSCF always had a minimum value at both ends of the elliptical major axis. With variation in the angle, the closer to both ends of the elliptical minor axis, the larger the value of αηDSCF , and the maximum value was obtained at both ends of the elliptical minor axis. Distribution of αηDSCF was symmetric about x -axis. In different cases, the minimum value ofαηDSCF was 0, but the maximum value of αηDSCF was different. The maximum value of αηDSCF in Fig. 4 was only 0.781, 0.662, 0.503 and 0.384, respectively. The maximum value of αηDSCF in Fig. 5 was 1.105, 1.202, 1.412 and 1.646, respectively. The maximum value ofαηDSCF in Fig. 6 was 1.136, 1.284, 2.079, and 1.965, respectively. The value ofαηDSCF in case 1 was less than 1, which meant that when the inclusion was stiffer than the medium, the steady-state incidence reduced angular stress dynamic concentration around inclusion. The value ofαηDSCF in cases 2 and 3 was larger than 1, which indicated that when inclusion was softer than medium, steady-state incidence aggravated the angular stress concentration around inclusion. The value ofαηDSCF in case 3 exceeded that in case 2, which revealed that the softer the inclusion was than the medium, the more significant the angular stress concentration around inclusion was. In addition, the distribution ofαηDSCF also changed with the axial ratio. As shown in Fig. 4 (d), Fig 5. (d), Figs. 6 (c) and (d), as the shape of inclusion was gradually close to circle and the wave number was high, αηDSCF exhibited multiple extreme values. Because under the condition of high wave number, there were multiple stress wave crests in the inclusion, and the positions of crests were different due to the influence of the phase difference, resulting in multiple extreme values.
Figure 7 showed the changes in the steady-stateαηDSCF with the radial coordinates and wave number when η =90°in three cases. As shown in Fig. 7(a), under the condition of constant wave number,αηDSCF of case 1 decreased slowly with an increase in the radial coordinate, and the value ofαηDSCF was always less than 1. The results demonstrated that as the inclusion was stiffer than the medium, the steady-state incidence diminished the angular stress concentration, and the more the shape of inclusion was closer to circle, the greater the degree of reduction. In Figs. 7(c) and (e), under the condition of constant wave number,αηDSCF of case 2 and case 3 had a certain volatility as the radial coordinate changes, and had a trend of oscillation. The oscillation at k 1=1 was more intense than that at k 1=0.5. Moreover, oscillation of αηDSCF in case 3 had a higher amplitude, a shorter period, and more intense than that in case 2. This implied that for the inclusion softer than the medium, it had a high sensitivity to radial coordinates, and the greater the difference between the material properties of the inclusion and medium, the higher the sensitivity.
As shown in Fig. 7(b), under the condition of constant elliptical axial ratio, αηDSCF of case 1 decreased with an increase in the wave number, eventually approached 0, and the value of αηDSCF was always less than 1. This meant that for the inclusion was stiffer than the medium, the high wave number steady-state incidence decreased the angular stress concentration. In Figs. 7(d) and (f), under the condition of constant elliptical axial ratio, theαηDSCF of cases 2 and 3 experienced obvious volatility with the variation in wave number, along with had a trend of oscillation. The oscillation at ξ=1.5 was more intense than that at ξ=0.5. In addition, oscillation ofαηDSCF in case 3 was higher in amplitude, shorter in period, and more severe than in case 2, which was consistent with the changes ofαηDSCF with the radial coordinates. The results indicated that for the inclusion was softer than medium, αηDSCF also had a high sensitivity to wave number. The greater the difference between the material properties of inclusion and medium, the higher the sensitivity. The phenomenon was consistent with the existing literature, which further confirmed the validity of the theoretical derivation.26