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.