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.