## 2 Theoretical Analysis of the Caustic Curve  
The caustic method is first proposed by [PERSON] [17], and then extended by [PERSON] [18], [PERSON] [19] and [PERSON] [20] to study the dynamic fracture problems under various loadings. Afterwards, [PERSON] [21] further studied the dynamic fracture problems associated with blast loading using the caustic technique. [PERSON] [22] developed the digital laser dynamic caustic system using high-speed camera and laser light source. Since then, the digital laser dynamic caustic system is widely used for studying the dynamic fracture mechanism of rock under blasting [8; 23; 24]. Figure 1 shows the formation of the caustic curve. When the parallel light goes through the stressed specimen, due to the significant changes in both the thick and refractive index of the specimen under blast loading, the parallel light could deviate from its original direction around the crack tip and eventually form a three-dimensional caustic envelope surface behind the specimen, as shown in Figure 1a. Therefore, a caustic curve can be generated behind the specimen, as shown in the red curve in Figure 1b under the tensile-shear stress condition. The black area surrounded by the caustic curve is the caustic spot. Based on the geometric relation of light deflection in Figure 1, the relation between the point A(x,y) on the specimen and the point A(x,y') on the reference plane can be expressed as [19]:  
\[\overset{\rightarrow}{X}\ =\ \overset{\rightarrow}{x}+\overset{\rightarrow}{w} \tag{1}\]  
where \(\overset{\rightarrow}{w}\) is the deviation vector and can be determined as follows:  
\[\overset{\rightarrow}{w}\ =\ -z_{0}\text{grad}\Delta S(r,\varphi) \tag{2}\]where \(\rm Z_{0}\) is the distance between the specimen and the reference plane; \(\rm\Delta S(r,\varphi)\) is the change of the optical path length, which can be given as:  
\[\rm\Delta S(r,\varphi)\ =\ (n_{0}-n_{a})\Delta d+d\rm\Delta n_{i} \tag{3}\]  
Here, \(\rm n_{0}\) is the initial refractive index of the specimen; \(\rm n_{a}\) is the refractive index of the atmosphere, and is equal to 1; \(d\) is the thickness of the specimen for a transparent plate; \(\Delta d\) is the change in thickness of the specimen under blast load; and \(\rm\Delta n_{i}\) is the change in the refractive index of the specimen related to the principal stress. For a plate in plane stress condition, \(\rm\Delta n_{i}\) can be expressed as:  
\[\left\{ \begin{array}{l}\rm\Delta n_{1}\rm=A\sigma_{1}\rm+B\sigma_{2}\\ \rm\Delta n_{2}\rm=A\sigma_{2}\rm+B\sigma_{1}\end{array} \right. \tag{4}\]  
where A and B are the material constants and are equal to \(\rm 0.53\times 10^{-10}\) m\({}^{2}\)/N and \(\rm-0.57\times 10^{-10}\) m\({}^{2}\)/N, respectively, for polymethyl methacrylate (PMMA) plate. \(\rm\sigma_{1}\) and \(\rm\sigma_{2}\) are the maximum and minimum principal stress, respectively. According to the caustic theory, the mapping equations of the caustic curve on the reference plane can be expressed as:  
\[\left\{ \begin{array}{l}x\prime\ =\ r_{0}\rm\left[\cos\phi+\frac{2}{3\sqrt{k_{1}^ {2}+k_{\rm II}^{2}}}\rm\left(K_{\rm I}\cos\frac{3\phi}{2}-K_{\rm II}\sin\frac{ 3\phi}{2}\right)\right]\\ y\prime\ =\ r_{0}\rm\left[\sin\phi+\frac{2}{3\sqrt{k_{1}^{2}+k_{\rm II}^{2}}}\rm \left(K_{\rm I}\sin\frac{3\phi}{2}+K_{\rm II}\cos\frac{3\phi}{2}\right)\right] \end{array} \right. \tag{5}\]  
where \(\rm r_{0}\) is the initial curve on the specimen, that forms the caustic curve on the reference plane, and can be determined as:  
\[r_{0}\ =\ \rm\left(\frac{3z_{0}d\rm c}{2\sqrt{2\pi}}\rm\right)^{2/5}\rm\left(K _{\rm I}^{2}+K_{\rm II}^{2}\right)^{1/5} \tag{6}\]  
In Equation (6), \(c\) is the stress-optical constant for the material and is equal to \(\rm 0.85\times 10^{-10}\) m\({}^{2}\)/N for PMMA. Therefore, the dynamic stress intensity factors \(\rm K_{\rm I}^{d}\) and \(\rm K_{\rm II}^{d}\) of the crack tip can be expressed as:  
\[\left\{