## 4 Anisotropy Analysis
### Mechanism of Compressive Strength Anisotropy Based on Maximum Critical Tangential Stress  
This study systematically elucidates the intrinsic mechanism by which pore defects induce anisotropic compressive strength in 3 DPC, based on variations in critical tangential stress (\(\sigma_{\theta c}\)) across different loading directions. It should be noted that flexural strength and interfacial splitting tensile strength are primarily governed by interfacial bond strength and porosity, with their mechanisms being relatively well established and thoroughly discussed in Section 3.3, and thus will not be reiterated here. Under vertical compressive stress (where \(\sigma_{1}\) = 0 and \(\alpha\) = 0\({}^{\circ}\)), for comparative purposes, we set \(\sigma_{2}\) = 1 MPa and \(r\) = 0.05 mm. Under these conditions, the expression for \(\sigma_{\theta c}\) can be simplified to  
\[\sigma_{\theta c}=-\frac{1}{2\sqrt{0.2\pi}}\left(\sqrt{\frac{\pi(d^{2}+1)(d^{2 }-1)(a+b)^{2}}{d[(d^{2}-1)a+(d^{2}+1)b]}}\right) \tag{9}\]  
In Figure 10, the stress convention follows compression-positive and tension-negative notation. Equation (9) yields \(\sigma_{\theta c}<0\), indicating tensile stress concentration that aligns with the concrete's characteristic brittle tensile failure mode. To investigate pore geometry effects, we maintained a constant crack length (\(c\) = 0.1 mm) and minor axis (\(b\) = 1 mm) while systematically varying the major axis (\(a\) = 0.1-10 mm), obtaining the \(\sigma_{\theta c}\) - \(a\) relationship shown in Figure 11. The results demonstrate that the tensile \(\sigma_{\theta_{c}}\) decreases with increasing \(a\), indicating that elliptical pores with more elongation along the load direction exhibit reduced the maximum tensile force at impending crack initiation. Consequently, such pore geometries require higher stress levels to reach the ultimate tensile strength and initiate crack formation, thereby enhancing the specimen's resistance to failure. Based on the theoretical analysis in Section 4.1, the aspect ratios of ellipsoidal pores vary significantly under different loading directions, as shown in Figure 12. Cross-sectional observations reveal distinct pore morphology characteristics across orientations: The Z direction exhibits the most flattened pores, followed by the Y direction, while the X direction shows minimal flattening. This geometric anisotropy directly governs the magnitude of critical tangential stress \(\sigma_{\theta_{c}}\). Under Z direction compression, \(\sigma_{\theta_{c}}\) reaches its maximum value, most readily attaining the concrete's tensile strength and initiating failure. In contrast, X direction loading, where pore long axes align with the loading direction, demonstrates the weakest stress concentration and the lowest \(\sigma_{\theta_{c}}\), resulting in optimal load-bearing capacity. The Y direction displays intermediate \(\sigma_{\theta_{c}}\) values, with compressive strength lower than the X direction but higher than the Z direction. In addition, failure modes further differ between orientations: Y direction failure primarily involves interlayer sliding, whereas Z direction failure is dominated by intralayer sliding [23]. Although both originate from interfacial weak zones formed during printing, the gravitational compaction of upper layers enhances interlayer bonding strength compared to intralayer interfaces. Consequently, the Y direction requires greater loading to induce interfacial sliding failure than the Z direction under compression. Notably, increased flowability in fresh concrete significantly improves material homogeneity. As demonstrated in Section 3.1, higher flowability promotes transformation  
Figure 11: The variation curve of \(\sigma_{\theta_{c}}\) with the value of \(a\). Figure 12: Cross-sectional views of elliptical pores under compressive loading in different directions. of elliptical pores toward more spherical geometries with reduced flattening, while simultaneously enhancing interfacial packing density and decreasing large pore content.