3.1 Case study and verification
In the elliptical coordinate system, the major and minor axis and axis ratio of any ellipse can be calculated by Eq (3). To simplify the problem, the focal length of the ellipse is taken as a =1m. The four cases of radial coordinate ξ = 0.1, 0.2, 0.5, 1.5 are determined to study the influence of different axial ratio on the scattering of incident SH wave around the elliptical inclusion. The smaller ξ , the larger the elliptical axis ratio and the closer the shape is to crack. The larger ξ , the smaller the elliptical axis ratio, and the closer the shape is to circle. The parameter settings of the radial coordinate are illustrated in Table 1.
In this study, the wave velocity cs was set as 2300 m/s, the incident wave numbers (k ) were predetermined to be 0.2, 0.5 and 1, respectively. The range of stress wave numbers generated by earthquake, engineering blasting and mostly impacts was covered here in.14 The difference in the material properties of the medium and elliptical inclusion additionally affected the scattering of incident SH wave around the elliptical inclusion.26This study set up three cases for calculation, as shown in Table 2.
In Table 2,k* =k 2/k 1,μ* =μ 1/μ 2 and three cases correspond to the inclusion being stiffer, softer and much softer than the medium, respectively.
Subsequently, the case was computed when the material properties parameters of the inclusion and medium were the same to verify the correctness of the derivation. When the inclusion and medium possessed the identical material properties parameters, the propagation of SH wave in the inclusion was the same as the propagation in the medium, and dynamic stress concentration was only related to the phase difference in the stress wave. Therefore, incident SH wave only produced the incident wave, and not generated the scattered and standing waves, leading to αDSCF determined only by the incident wave. According to the definition ofαηDSCF , the maximum value ofαηDSCF was 1. The numerical results are shown in Fig. 3.
In Fig. 3, as the incident wave number and axial ratio changed,αηDSCF had a maximum value at the angle perpendicular to the incident direction (both ends of the elliptical minor axis), and a minimum value at the angle of incidence (both ends of the elliptical major axis). The maximum and minimum values of αηDSCF were 1 and 0.αηDSCF gradually increased with the angle from 0° to 90°, and distribution ofαηDSCF was symmetrical about the x and y axes. To verify the correctness of the theoretical derivation, the numerical results were compared with those available in literature. As expected, the verification exhibited excellent agreement between the results of the present study and those available in the literature.10