# Transient Response of Dynamic Stress Concentration around a Circular Opening: Incident SH Wave
## 4 Dynamic Stress Concentration Coefficient and Dynamic Stress Coefficient  
In the transient problem, the range of the wave field is considered to be limited only to the vicinity of the circular opening, so that the surrounding boundary can be ignored. The boundary conditions for this problem are  
\[\tau_{\tau_{2}}|_{r=r_{0}}=0, \tag{21}\]  
where \(r_{0}\) is the radii of the opening.  
The infinite series \(\tau_{\tau z}\) is identical to zero in the time period \(t\in[0,\mathrm{T}+2r_{0}/c]\), requiring every term in the series to be identical to zero, and since \(\cos\!m\theta\) is orthogonal for all \(\mathrm{m}\) in the interval \((0,2\pi)\), the coefficient of \(\cos\!m\theta\) must be zero. Upon substituting Equation (19) into Equation (21) to obtain undetermined coefficients, the result is:  
\[A_{m}=-\frac{\mathrm{i}^{m}\varepsilon_{m}\big{[}\int_{m-1}\big{(}\frac{m \omega_{0}r_{0}}{c}\big{)}-J_{m+1}\big{(}\frac{m\omega_{0}r_{0}}{c}\big{)} \big{]}}{H_{m-1}^{(1)}\big{(}\frac{m\omega_{0}r_{0}}{c}\big{)}-H_{m+1}^{(1)} \big{(}\frac{m\omega_{0}r_{0}}{c}\big{)}}. \tag{22}\]  
The dynamic stress of the circular opening can be expressed as the modulus of the imaginary part of the solution, with the following result:  
\[\tau_{\theta_{2}}^{*}|_{r=r_{0}}=|\mathrm{Im}\tau_{\tau z}|=\frac{h}{r_{0}} \prod(t)\bigg{|}\sum_{n=1}^{\infty}\mathrm{Im}\tau_{\theta z}^{(n)}\bigg{|}. \tag{23}\]  
The dynamic stress component of the elastic field can be expressed as a separation of time and space variables, with the result:  
\[\tau_{\theta z}^{(n)}(r,\theta,t^{\prime})=\tau_{\theta z}^{(n)}(r,\theta)T^{ (n)}\big{(}t^{\prime}\big{)}, \tag{24}\]  
where \(\tau_{\theta z}^{(n)}(r,\theta)\) is a space function and \(T^{(n)}(t^{\prime})\) is a time function, and the imaginary part of their product is the actual dynamic stress.  
By substituting Equation (17) into Equation (24), we obtain:  
\[\tau_{\theta z}^{(n)}(r,\theta)=\Big{[}\mathrm{i}^{m}\varepsilon_{m}J_{m}( \frac{n\omega_{0}r}{c})+A_{m}H_{m}^{(1)}(\frac{n\omega_{0}r}{c})\Big{]}m\sin m\theta, \tag{25}\]  
\[T^{(n)}\big{(}t^{\prime}\big{)}=a_{n}\mathrm{e}^{-\mathrm{i}m\omega_{0}t^{ \prime}}. \tag{26}\]  
The space function \(\tau_{\theta z}^{(n)}(r,\theta)\) can be written in the following form:  
\[\tau_{\theta z}^{(n)}(r,\theta)=C_{n}+\mathrm{i}D_{n}, \tag{27}\]where \(C_{n}\) and \(D_{n}\) are the real and imaginary parts of \(\tau_{\theta_{2}}^{(n)}(r,\theta)\).  
Substituting Equations (26) and (27) into Equation (23), we can obtain:  
\[\tau_{\theta_{2}}^{*}\big{(}r,\theta,t^{\prime}\big{)}\big{|}_{r=\tau_{0}}=\frac {\mu}{r_{0}}\prod(t)\left|\sum_{n=1}^{\infty}E_{n}(\theta)\sin\big{[}n\omega_{ 0}t^{\prime}-\phi_{n}(\theta)\big{]}\right|, \tag{28}\]  
where \(E_{n}(\theta)=-a_{n}\sqrt{C_{n}^{2}+D_{n}^{2}}\), \(\phi_{n}(\theta)=\arctan(C_{n}/D_{n})\).  
Using complex variable \(\zeta=\theta+\mathrm{i}t\), \(\overline{\zeta}=\theta+\mathrm{i}t\), assume that the opening edge stress is  
\[\tau_{\theta_{2}}^{*}\big{(}r_{0},\theta_{j},t_{j}\big{)}=\frac{\mu}{r_{0}} \prod(t_{j})\sum_{n=1}^{\infty}\big{|}E_{n}\big{(}\theta_{j}\big{)}\sin\big{[} n\omega_{0}t_{j}-\phi_{n}(\theta_{j})\big{]}\big{|}. \tag{29}\]  
The DSCC can then be defined as [14]:  
\[DSCC=\bigg{|}\frac{\tau_{\theta_{2}}^{*}(\zeta)}{\tau_{0}}\bigg{|}, \tag{30}\]  
where \(\tau_{0}=\max\Big{(}\tau_{\theta_{2}}^{(i)}\Big{)}\) represents the maximum stress generated by incident wave \(\tau_{\theta_{2}}^{(i)}\).