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