Blood clots represent living materials composed of a polymer network and an abundance of cells. They might fracture within the bulk material of the clot (cohesive fracture), at the interface between the clot and the surrounding tissue (adhesive fracture), or through a combination of both modes (hybrid fracture). The clot fracture within vascular systems and injury sites could lead to life-threatening conditions. Despite the significance, understanding and modeling the fracture behaviors of blood clots, including their dependence on mechanical loading and cellular components, remain in a nascent stage. In this study, we employ an integrated experimental-computational approach to comprehensively investigate the fracture behaviors of bovine blood clots. We explore various mechanical factors, substrates, and cellular components such as red blood cells (RBCs) and platelets. Our findings reveal that among various tissue substrates, blood clots exhibit the highest interfacial adhesion energy with muscle, and the lowest to the inner arterial lining, consistent with their biological function. Both interfacial adhesion energy and bulk fracture energy are rate-dependent, although they exhibit different dependencies. Also, RBCs and platelets have different effects on clot fracture. An increase in RBC content tends to toughen both adhesion and fracture of blood clots. However, an increase in platelet content enhances interfacial adhesion energy but lowers the bulk fracture energy. The platelet content also governs the shift from adhesive fracture to hybrid fracture. To model clot fracture, we developed two finite element models incorporating a coupled cohesive-zone and Mullins-effect approach to simulate pure shear fracture and peeling of blood clots. These models, validated through experimental data, elucidate the interplay between intrinsic fracture toughness, interfacial strength, and bulk energy dissipation during clot fracture. This study significantly advances our understanding of clot mechanics, providing valuable insights into the mechanics of similar living materials and the management of clot-related disorders such as hemorrhage and thrombosis.