3 Steady-state response of inclusion
The scattering wave function
generated around the elliptical inclusion satisfies the two-dimensional
stable wave equation and Sommerfeld’s radiation condition. Therefore,
according to the asymptotic characteristics of the Mathieu function, the
scattering wave φs can be expressed as:
where Bm and Cm denote
undetermined coefficients for satisfying the boundary conditions, andq =(ak )2/4.
In accordance with the principle of wave superposition, the full wave
function in the medium can be expressed as:
In addition, the standing wave generated in the elliptical inclusion,
can be represented as10:
In the elliptical coordinate system, the stress component is given by
the following formula25:
The undetermined coefficient can be determined by applying the boundary
conditions. The boundary conditions require continuity of the
displacement and stress along the inclusion surface, which can be
expressed as:
When the incident wave is parallel to the x -axis, θ=0,
sem(0, q) =0, Eqs (13), (14) and (16) can be simplified
as:
Substituting Eqs (19), (20) and (21) into Eq (18) to derive the
following two equations:
cem(η,q 1) andcem(η,q 2) are not
orthogonal to each other, so the undetermined coefficientsBm and Dm cannot be
calculated directly. Thus, the orthogonal condition of angular Mathieu
function is used to simplify the calculation. The orthogonal condition
can be expressed as:
Multipling both ends of Eq (22) by cen(η,
q 1) and integrating from 0 to 2π to derive the
following two simplified equations:
where
Eliminating Bm from the two equations of Eq (24)
to determine an algebraic equation system aboutDm .
where
Based on the characteristics of the Mathieu function, it can become a
finite series by truncating from N term, and the numerical approximate
solution can be obtained. Taking N equations in Eq (26) to compute the
coefficients D 0, D 1,D 2, etc. Then bringing back Eq (24) to obtainBn , which is expressed as follows:
Therefore, the full wave function in the medium can be expressed as:
The dynamic stress concentration factor αDSCFaround the elliptical inclusion is defined as the ratio of the stress
produced by the full wave and the peak stress produced by the incident
wave, and it can be expressed as:
Substituting Eq (29) into Eq (30) to obtain the steady-state angular
dynamic stress concentration factorαηDSCF and steady-state radial
dynamic stress concentration factorαξDSCF, the results are
represented as follows:
Owing to the characteristics of harmonic function, only the real or
imaginary part of the result of Eq (31) represents the
steady-state-state αDSCF . Adding the
time-dependent term e-iωt , the real part
represents steady-state αDSCF at T=0, and the
imaginary part represents steady-state αDSCF at
T/4. T denotes the period of the incident wave.