## 1 Introduction  
Rocks with existing cracks are prevalent in natural formations and are encountered in various engineering applications, such as tunneling, mining, and the construction of underground structures [1, 2]. The presence of cracks influences the overall strength, deformability, and failure mechanisms of rock masses [3, 4, 5]. The exploration of the fracture behavior and mechanical response of cracked rock masses not only holds significance for understanding the structural integrity and stability of geological formations, but also plays a pivotal role in enhancing the safety and efficiency of engineering projects. By subjecting rock specimens to controlled laboratory tests, researchers gain valuable insights into the fundamental properties governing the strength, deformability, and failure mechanisms of cracked rocks. [PERSON] and [PERSON] [6] focused on the analysis of damage process of rocks that contain some pre-existing fractures. [PERSON] et al. [7] investigated the impact of the crack dip angle on the mechanical properties and failure modes of pre-cracked red sandstone in underground engineering, revealing a five-stage stress-strain curve and an increase in peak strength with the crack dip angle. [PERSON] et al. [8] conducted uniaxial compression and acoustic emission tests on sandstones with different joint dip angles, revealing a shift in the failure mechanism from tensile to shear with a critical angle of 45\({}^{\circ}\). [PERSON] et al. [9] studied the influence of crack dip angles on the energy characteristics of sandstones, showing that larger crack dip angles correspond to a higher energy storage but lower energy dissipation in rocks, explaining the enhanced bearing capacity with larger crack dip angles from an energy perspective. [PERSON] et al. [10] and [PERSON] et al. [11] explored the effect of crack dip angles on the mechanical properties and fracture evolution mechanism of granite under uniaxial and triaxial compression, revealing weakened brittle and enhanced plastic characteristics with varying dip angles, impacting compressive strength, elastic modulus, and failure modes. These studies not only enhance the understanding of the mechanical properties of fractured rocks, but also provide a theoretical basis for geotechnical engineering, structural design, and geological hazard assessments. By combining laboratory experiments with advanced numerical modeling techniques, researchers have conducted in-depth research on microcrack initiation, propagation, and coalescence at the tip of joints, seeking to obtain the basic mechanical response and fracture characteristics of cracked rocks under different loading conditions. In terms of numerical simulation methods, cracked rock simulation is mainly achieved through the extended finite element method (XFEM) [12; 13], the discrete element method (DEM) [14; 15], the phase-field method [16; 17], the finite-discrete element method (FDEM) [18; 19], and the peridynamics (PD) [20]. Compared with laboratory tests, numerical simulations can provide information on the stress distribution evolution and microcrack propagation path during the rock fracture process [21; 22; 23; 24]. This study analyzed the deformation and mechanical characteristics of granite specimens containing pre-existing cracks during loading through uniaxial compression tests and numerical simulations. The numerical simulation was based on the cohesive element method, which uses one or more cohesive elements to model crack behavior. These elements introduce a physical parameter known as \"cohesive\" or \"adhesive strength\" near the crack surfaces. These cohesive forces simulate the interaction between materials along the crack or contact interface, allowing for the precise representation of crack initiation, propagation, and closure in numerical simulations.