# Transient Response of Dynamic Stress Concentration around a Circular Opening: Incident SH Wave
## 3 Incident Waves and Scattered Waves  
Assuming that the generation time of the SH wave is T, the wave field of the incident wave at time interval \(t\leq 0\) is represented as \(w^{(i)}=0\). Therefore, based on the properties of the incident wave, the wave field function is transformed into an odd function with a period of 2T using odd extension. As shown in Figure 1, the SH wave is incident from the right. Furthermore, by expanding the Fourier series, the wave field of the incident wave can be expressed as the superposition of sine waves with different frequencies and amplitudes, with the result:  
\[w^{(i)}=\sum\limits_{n=1}^{\infty}a_{n}\sin\Bigl{(}n\omega_{0}\frac{r}{c}\cos \theta-n\omega_{0}t\Bigr{)}, \tag{6}\]  
where \(a_{n}\) is the harmonic amplitude coefficient, \(\omega_{0}=\pi/\mathrm{T}\) is the fundamental frequency, and \(t\) is the wave field response time.  
Equation (6) is rewritten in complex exponential form as [11]:  
\[w^{(i)}=\sum\limits_{n=0}^{\infty}a_{n}\mathrm{e}^{\mathrm{i}(n\omega_{0}\frac {r}{c}\cos\theta-n\omega_{0}t)}, \tag{7}\]  
where the imaginary part of \(w^{(i)}\) represents the actual displacement of the particle.  
Figure 1: Schematic diagram of SH wave incident.  
The Fourier transform of Equation (7) can be expressed as:  
\[\varphi^{(i)}(r,\theta,\omega)=2\pi\sum\limits_{n=1}^{\infty}a_{n}\delta(\omega+n \omega_{0})\mathrm{e}^{-\mathrm{i}kr\cos\theta}. \tag{8}\]  
Using the wave function expansion method, Equation (7) is expressed as [4]:  
\[w^{(i)}=\sum\limits_{n=1}^{\infty}\left[a_{n}\mathrm{e}^{-\mathrm{i}n\omega_{0 }t}\sum\limits_{m=0}^{+\infty}\mathrm{i}^{m}\varepsilon_{m}J_{m}(\frac{n \omega_{0}r}{c})\cos m\theta\right]. \tag{9}\]  
The wave field of the scattered wave generated by the circular opening can be written as [4]:  
\[w^{(s)}=\sum\limits_{n=1}^{\infty}\left[a_{n}\mathrm{e}^{-\mathrm{i}n\omega_{0 }t}\sum\limits_{m=0}^{+\infty}A_{m}H_{m}^{(1)}(\frac{n\omega_{0}r}{c})\cos m \theta\right], \tag{10}\]  
where \(J_{m}(\cdot)\) represents the first kind Bessel function, and \(H_{m}^{(1)}(\cdot)\) represents the first kind Hankel function.  
Then, the total elastic field can be expressed as:  
\[w^{(i)}=w^{(i)}+w^{(s)}. \tag{11}\]  
By substituting Equations (9) and (10) into Equation (2), the shear stress generated by the wave field of the incident wave can be obtained, and the result is:  
\[\tau_{rz}^{(i)}=\mu\sum\limits_{n=1}^{\infty}\frac{n\omega_{0}}{2c}\Bigg{\{}a_ {n}\mathrm{e}^{-\mathrm{i}n\omega_{0}t}\sum\limits_{m=0}^{\infty}A_{m}\Big{[}J _{m-1}(\frac{n\omega_{0}r}{c})-J_{m+1}(\frac{n\omega_{0}r}{c})\Big{]}\cos m \theta\Bigg{\}}, \tag{12}\]  
\[\tau_{\theta_{2}}^{(i)}=\frac{\mu}{r}\sum\limits_{n=1}^{\infty}\left[a_{n} \mathrm{e}^{-\mathrm{i}n\omega_{0}t}\sum\limits_{m=0}^{\infty}A_{m}H_{m}^{(1 )}(\frac{n\omega_{0}r}{c})m\sin m\theta\right]. \tag{13}\]  
Simultaneously, the scattered wave generated by an irregular opening can be expressed as:  
\[\tau_{rz}^{(s)}=\mu\sum\limits_{n=1}^{\infty}\frac{n\omega_{0}}{2c}\Bigg{\{}a _{n}\mathrm{e}^{-\mathrm{i}n\omega_{0}t}\sum\limits_{m=0}^{\infty}A_{m}\Big{[} H_{m-1}^{(1)}(\frac{n\omega_{0}r}{c})-H_{m+1}^{(1)}(\frac{n\omega_{0}r}{c}) \  
The relation between the wave field propagation time and the dynamic stress response time around the circular opening is:  
\[t^{\prime}=t-s/c, \tag{18}\]  
where \(t^{\prime}\) is the response time of the dynamic stress and \(s\) is the vertical distance from the incident position of the wave field to the edge of the circular opening.  
Then, the dynamic stress of the circular opening can be expressed by the result after delay, and the result is:  
\[\tau_{\sigma_{2}}\big{(}t^{\prime}\big{)}=\tau_{\sigma_{2}}^{(t)}\big{(}t^{ \prime}\big{)}\prod(t), \tag{19}\]  
\[\tau_{\theta_{2}}\big{(}t^{\prime}\big{)}=\tau_{\theta_{2}}^{(t)}\big{(}t^{ \prime}\big{)}\prod(t), \tag{20}\]  
where the rectangle function \(\prod(t)=h(t-a)-h(t-a-T)\), \(a=\frac{s+r_{0}(1+\cos\theta)}{c}\). \(h(\cdot)\) is the unit step function and \(t\in[0,\mathrm{T}+(s+2r_{0})/c]\), \(t_{0}=a+\mathrm{T}/2\).