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The study tackles the challenge of accurately modeling fracture behavior in beam lattices, which is essential for designing robust architected materials. Our research focuses on evaluating how the lattice's microstructure and material properties affect fracture toughness. We employed finite element simulations based on the Euler-Bernoulli beam theory to investigate crack propagation, using a failure criterion that initiates beam fracture when maximum axial stress exceeds critical strength. Building on observations from these simulations, we developed a multi-phase-field fracture model with Cosserat elasticity to integrate consistent toughness characteristics into a comprehensive framework for lattice design. This model was validated through experimental tests, ensuring a close match between theoretical predictions and physical reality. Our findings reveal that the energy release rate remains relatively stable during crack propagation, underscoring its reliability as a measure of the toughness of periodic lattice structures. We discovered that toughness is predominantly influenced by beam height and material properties such as tensile strength and Young's modulus, while slenderness has minimal impact. Additionally, cracks were observed to preferentially propagate along the lattice's structural directions due to stress localization effects, highlighting the importance of the microstructure in fracture behavior. The implications of this research are significant, suggesting that improved modeling of fracture in lattice structures can enhance material design reliability and optimization. This study bridges the gap between theoretical models and real-world applications, providing valuable insights for the development of advanced materials with tailored fracture properties.