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: