# Dynamically Triggered Damage Around Rock Tunnels: An Experimental and Theoretical Investigation
## 4 Stress State Around the Arched Hole Under Dynamic Impact  
### Conformal Mapping of Arched Hole  
As mentioned, the damage at the periphery of an arched hole should be considered a two-dimensional stress wave scattering problem. The parallel orientation of the upper and lower surfaces of the specimen with respect to the direction of impact eliminates wave reflection upon incidence. Reflection and transmission occurring at the interfaces at both ends have no effect within the observation period. Consequently, the problem can be simplified to the scattering of stress waves in the vicinity of an arched cavity within a fully planar space. In order to understand the mechanism behind this damage, it is crucial to determine the stress distribution around the hole, as stress is the primary factor influencing material failure. Typically, for a circular hole subjected to static pressure, the stress distribution at the hole can be obtained using Kirsch's solution [33]. However, when the hole is subjected to dynamic loading, the stress distribution is obtained through the wave function expansion method. Analytically obtaining the stress distribution around a complex boundary is extremely challenging. Therefore, it is necessary to transform the complex boundary into a unit circle using complex functions and the conformal transformation method as shown in Figure 8. This transformation can be expressed in the form of a Laurent series [34]:  
\[z=w(\zeta)=R\left(\zeta+\sum\limits_{k=0}^{\infty}C_{k}\zeta^{-k}\right),\ \ \ |\zeta|\leq 1 \tag{2}\]  
Here, \(w(\zeta)\) is a holomorphic function, \(z\) and \(\zeta\) are complex variables, \(C_{k}\) is the Laurent expansion coefficient, and \(R\) is the shape-dependent scale factor. Expanding Equation (2) in the complex plane gives:  
\[r_{j}e^{i\alpha_{j}}=R\left(e^{i\beta_{j}}+\sum\limits_{k=0}^{\infty}C_{k}e^{- i\beta_{j}}\right) \tag{3}\]  
Here, \(\theta_{j}\) and \(\alpha_{j}\) are the corresponding polar angles in the physical and image planes, and \(\eta_{j}\) is the polar diameter in the physical plane.  
Applying [PERSON]'s formula to Equation (3) and separating the real and imaginary parts, the following two equations can be obtained:  
\[\left\{\begin{array}{l}\sin\left(\alpha_{j}-\theta_{j}\right)+\sum\limits_{ k=0}^{\infty}C_{k}\sin\left(\alpha_{j}+k\theta_{j}\right)=0\\ r_{j}=R\left[\cos\left(\alpha_{j}-\theta_{j}\right)+\sum\limits_{k=0}^{\infty} C_{k}\cos\left(\alpha_{j}+k\theta_{j}\right)\right]\end{array}\right. \tag{4}\]  
\[R=r_{1}/\left(1+\sum\limits_{k=0}^{n-1}C_{k}\right) \tag{5}\]  
Figure 8: Mapping of arches to unit circle.  
The problem described in Equations (4) and (5) can be approached as a constrained optimization problem [35] to find an approximate expression for the finite term, aiming to minimize the error between the mapped shape and the original shape. Alternatively, the problem can be directly solved by truncating the series using an analytical method [36]. It is widely accepted that a satisfactory mapping shape can be achieved by expanding the Laurent series to include 6-7 terms. Specifically, for the arched shape shown in Figure 1, its corresponding mapping function can be derived as follows:  
\[w(\zeta)= \left\{\begin{array}{l}0.5424\zeta-0.059-0.0036\zeta^{-1}+0.0458 \zeta^{-2}\\ -0.0432\zeta^{-3}+0.018\zeta^{-4}+0.004\zeta^{-5}-0.0046\zeta^{-6}\end{array}\right. \tag{6}\]  
Figure 9 plots the mapped shape determined by Equation (6) and the original shape. It can be seen in the figure that the mapped shape is almost identical to the original shape, except at the arch foot, where the original shape is a right angle and the mapped shape is a rounded corner with a small chamfer, but it is in a match with the specimen drill holes in Figure 1, where the holes in the specimen are similarly rounded with a tiny rounded corner.