## 4 Anisotropy Analysis
### Fracture Theory of Ellipsoidal Pores Based on Stress Intensity Factor  
Under external loading, the stress distribution at pore boundaries is significantly influenced by their geometric morphology [42]. According to [PERSON]'s microcrack theory [45], when the aspect ratio (L/w) of a pore along the loading direction is small, stress concentration preferentially occurs at the boundary, inducing crack initiation. As illustrated  
Figure 8: \(I_{\text{a}}\) with different flowabilities: (**a**) compressive strength; (**b**) flexural strength. in Figure 9a, compressive stress concentration at the left and right tips of an elliptical pore may lead to local crushing, while tensile stress concentration at the upper and lower tips tends to initiate tensile cracking. Given that the tensile strength of concrete is substantially lower than its compressive strength, failure typically occurs via brittle tensile cracking. Once stress concentration triggers crack formation at a pore, the stress field redistributes, altering the mechanical state of adjacent pores and promoting continuous crack propagation [21] (Figure 9b). Furthermore, such pore-induced microcrack evolution ultimately governs the macroscopic mechanical behavior of the material [46]. The stress intensity factor (\(K\)) serves as a fundamental parameter for evaluating material fracture. When \(K\) reaches the material's fracture toughness (\(K_{c}\)), the fracture initiates and propagates until complete specimen failure occurs. This mechanism is equally applicable for analyzing concrete fracture behavior [47]. For ellipsoidal pores in concrete, theoretical analytical models have been developed to calculate the stress intensity factor at crack tips under complex stress states [48], with the mechanical schematic shown in Figure 10. In the figure, \(a\) represents the major axis of the elliptical pore, \(b\) denotes the minor axis of the elliptical pore, and \(c\) indicates the crack length at the tip of the elliptical pore. The far-field compressive stresses are designated as \(\sigma_{1}\) and \(\sigma_{2}\), while \(\alpha\) represents the angle between \(\sigma_{1}\) and the major axis. The crack initiation angle is \(\theta_{0}\), \(r\) is the initiation radius, and \(\sigma_{\theta}\) signifies the tangential stress at the fracture point. For calculation purposes, compressive stresses are defined as positive and tensile stresses as negative. Figure 10: Schematic of crack initiation at ellipsoidal pore tips [48]. Figure 9: (**a**) Stress concentrations in 3D ellipsoidal pores with axial ratios (L/w) of 0.5, 1, and 2 [42]. (**b**) Diagram of the continuous propagation of the crack at the pore boundary. The stress intensity factor is calculated using Equations (5) and (6):  
\[K_{\rm I}=\sqrt{\frac{\pi(d+1)^{2}(d^{2}-1)(a+b)^{2}}{4d\Big{[}(d^{2}-1)a+(d+1)^{ 2}b\Big{]}}}\bigg{[}-\frac{\sigma_{1}+\sigma_{2}}{2}+\frac{\sigma_{1}-\sigma_{2 }}{2}\cos 2\alpha\bigg{]} \tag{5}\]  
\[K_{\rm II}=\sqrt{\frac{\pi(d^{2}+1)(d^{2}-1)(a+b)^{2}}{4d[(d^{2}-1)a+(d^{2}+1) b]}}\bigg{[}\frac{\sigma_{1}-\sigma_{2}}{2}\sin 2\alpha\bigg{]} \tag{6}\]  
Here, \(K_{\rm I}\) and \(K_{\rm II}\) correspond to Mode I (opening) and Mode II (sliding) crack types, respectively, where \(d=\frac{c+a+\sqrt{\left(c+a\right)^{2}-a^{2}+b^{2}}}{a+b}\). The crack initiation angle is determined by Equation (7):  
\[\theta_{0}= \begin{cases}0,&K_{\rm II}=0\\ 2{\rm arctg}\left[\frac{1-\sqrt{1+8\left(\frac{K_{\rm II}}{K_{\rm I}}\right)^{ 2}}}{4\frac{K_{\rm II}}{K_{\rm II}}}\right],&K_{\rm II}\
eq 0\end{cases}  \tag{7}\]  
Based on this, the maximum tensile force at impending crack initiation can be calculated using Equation (8):  
\[\sigma_{\theta c}=\frac{1}{2\sqrt{2\pi r}}\cos\frac{\theta_{0}}{2}\left[K_{ \rm I}(1+\cos\theta_{0})-3K_{\rm II}\sin\theta_{0}\right] \tag{8}\]  
A fracture occurs when \(\sigma_{\theta c}\) exceeds the concrete's tensile strength \(\sigma_{t}\). As the cracks propagate, microcracks induced by adjacent pores interconnect, ultimately leading to macroscopic failure of the specimen.