2.1 Problem statement
The deep-buried inclusion has been regarded as a problem of an infinite medium.SH wave as a common stress wave, negatively impacted on the inclusion structure in the propagation process. An elliptic cylindrical inclusion embedded in a full-space is subjected to a plane incident SH-wave. The incident angleθ is the angle between the incident direction and the positive direction of the x -axis. The geometry of the model is presented in Fig. 1. The major and minor axis of the elliptical inclusion are denoted by l andh . Both the medium and inclusion are isotropic elastic materials. Subscript 1 represents the parameters related to the surrounding medium, and subscript 2 refers to the parameters related to the elliptical inclusion. Their material properties include shear modulus μ and wave number k . The shear modulus and wave number of the medium denote μ 1 and k 1, and the shear modulus and wave number of the elliptical inclusion stand forμ 2 and k 2.
2.2 Wave function in elliptical coordinate system
The scattering and dynamic stress concentration around the elliptical inclusion can be solved by the wave function expansion method in the elliptical coordinate system. The elliptical coordinate system is shown in Fig. 2.
The elliptical coordinate system consists of numerous confocal ellipses and hyperbolas with the focal length of 2a . The transformation from rectangular coordinate system (x ,y ) to elliptical coordinate system (ξ,η ) is defined by18:
where ξ and η are the radial and angular coordinates of the elliptical coordinate system, respectively.
The scale factor in the elliptical coordinate system can be expressed as:
Therefore, the major axis, minor axis and axis ratio of the ellipse can be represented as:
The Helmholtz equation is obtained by separating time variables from the wave equation. In the elliptical coordinate system, the Helmholtz equation can be written as10:
where φ denotes the potential function of SH wave,cs denotes the velocity of SH wave, kdenotes wave number of SH wave and k=ω/cs ,ω denotes the circular frequency.
By varying separation equationφ (ξ, η )=X (ξ )Y (η ), the Mathieu equation can be expressed as:
whereq =(ak )2/4, and Eq (5) is referred to as the radial and angular Mathieu equations, respectively. The radial and angular Mathieu functions are obtained by solving the Mathieu equation. In order to obtain the unique single-valued solutions, only those periodic solutions with π or 2π period are of interest to us. This requires that parameters b andq satisfy certain functional relations, which can be expressed asF (b ,q )=0. The equation is called as the characteristic equation, and b in the equation represents the characteristic value.
The angular Mathieu function can be expanded by the Fourier series as a series sum of sine and cosine functions as the following symbols22:
The radial Mathieu function is expressed by different types of cylinder functions as follows22:
where Arm ,Brm denote the expansion coefficient associated with q . In Eq (7),j =1,2,3,4,Zmj represents the j th type cylinder function of m-order, and the corresponding Mathieu function is termed as the j th type radial Mathieu function. Radial Mathieu function and angular Mathieu function differ in variable by only one imaginary unit.
For simplicity, assuming that the only non-zero displacement component of incident plane SH wave isuz(i), and its maximum displacement is u 0. θ is the incident angle and the analytic expression of the incident wave can be expressed as:
Substituting Eq (8) into Eq (1) upon omitting the time-dependent terme-iωt , the expression of incident wave can be simplified as:
The radial Mathieu and angular Mathieu functions have the following integral relation.23
Since eikaw is a periodic function of the variable θ , eikaw can be represented by the expanded form as10:
where
In this study, the amplitude of incident wave u 0is set as 1. Substituting Eq (12) into Eq (11), the incident wave function expressed by the Mathieu function is derived as24: