1 Introduction
Due to the discontinuity of structure, inclusions often exist in different objects, such as underground structure in the stratum, impurities in materials and so on. Scattering and dynamic stress concentration will occur around inclusions when subjected to stress wave1,2, which may cause the structural failure of inclusions. Thus, dynamic response of a medium with inclusions embedded in elastic waves should be considered seriously. Inclusions usually have irregular shapes, which squares and circles are difficult to directly apply to practical research. Ellipse can approach circle and crack with the change in the axis ratio, which exhibits strong flexibility and renders it more suitable for practical engineering. Therefore, the study of the dynamic stress concentration surrounding elliptical inclusions demonstrates an important engineering significance.
In recent decades, the scattering and dynamic stress concentration of stress waves have been extensively studied, the models, methods and wave properties have been developed more maturely.3-9 Pao and Mow10 first synthesized and calculated the existing models of the scattering and dynamic stress concentration, obtaining the dynamic stress concentration distribution of a series of models and its influencing factors using the wave function expansion method. Subsequently, investigations have mainly focused on two theoretical types: cavity and inclusion. Liu et al11employed the complex variable function to solve the dynamic stress concentration problem surrounding cavity of arbitrary shape in an infinite elastic plane, providing the computational results of the dynamic stress concentration around the cavities of circular, elliptical and horseshoe shape. Tao et al12 solved the dynamic stress concentration around a circular cavity under the transient P wave disturbance in an infinite homogeneous medium based on the complex variable function and Fourier transform, and the dynamic stress concentration distribution around the circular cavity was observed to affect the Poisson’s ratio, wave number and waveform. Li et al13 explored the application of the complex variable function to determine the scattering of a shallow-buried circular cavity under the transient P wave loads, and analyzed the effects of cavity depth, incident angle and position of wave peak on the dynamic stress concentration factor (αDSCF ) distribution. A Butterworth filter was designed to remove the jump points and achieve more reasonable transient response results. Tao et al14 investigated the utilization of the wave function expansion method based on the Mathieu function to solve the scattering and dynamic stress concentration surrounding the elliptical cavity produced owing to the transient SH wave in the infinite plane, and simulated the plastic deformation of the cavity using LS-DYNA to validate against the numerical result. Ghafarollahi and Shodja15 implemented the multipole expansion method to present an analytical treatment for the anti-plane scattering of SH-waves by an arbitrarily oriented elliptic cavity/crack is embedded near the interface between the exponentially graded and homogeneous half-spaces.
In addition, the scattering and dynamic stress concentration around inclusions have always been the focus of research too, and the boundary conditions of inclusions are more complicated than cavities. Manoogian and Lee16 proposed the weighted residual method to the problem of the diffraction and scattering of plane SH-waves by an underground inclusion in half-space, and determined the ground-motion of circular, elliptical and square inclusions. Moreover, the effect of shape and depth of inclusions, frequency and angle of incidence of the incidence wave in ground-motion amplification was analyzed. Yang et al17 utilized the Green’s function to solve the scattering far field solution of SH-wave by a movable rigid cylindrical interface inclusion, indicating that different combinations of medium parameters exhibited a great influence on the far-field solution. Lee and Amornwongpaibun18 employed the wave function expansion method in the elliptical coordinates and elliptical cosine half-range expansion method to offer an analytical solution to the problem of the scattering around the semi-elliptical hill on half-space, and found that the existence of an elliptical hill causes complicated effects on ground motion. Hei et al19 presented a universal approach of solving the dynamic stress concentration around a circular inclusion in two-dimensional inhomogeneous medium based on the complex function theory. The inhomogeneity of medium is considered in the calculation process, which expands the research of complex medium. Sheikhhassani and Dravinski20 derived a non-hypersingular boundary integral equations to compute the stresses and αDSCF by using a weak form of Helmholtz equation. And using the method to evaluate dynamic stress concentration for the multiple multilayered inclusions embedded in an elastic half-space subjected to SH-waves. Qi et al21 suggested the way of the complex variable function method, combined with ”conformal mapping” method and Green’s function method to study the scattering problems of SH-wave by elliptical inclusion with partial debond curve and circular cavity in half-space. Results revealed thatαDSCF was influenced by the incident angle, the frequency of incident wave, distance between the defects, depth of the inclusions and partial debond curve angle.
However, the existing research has mainly focused on studying the steady-state stress response and angular dynamic stress concentration around elliptical inclusions while few studies have been devoted to the transient response of stress wave and radial dynamic stress concentration. Thus, it is vital to further extend the investigation of the radial stress concentration distribution around the elliptical inclusions and the dynamic response caused by transient incidence. In this study, theoretical solutions based on the wave function expansion method and Fourier transform were developed for an inclusion in infinite space when subjected to a plane SH-wave. The steady-state and transient responses of the radial and angular stress were determined, and the effects of wave numbers, elliptical axial ratio and material properties on dynamic response were analyzed and discussed.