## 1 Introduction  
As a typical geomaterial, rock mass is characterized by non-homogeneity, discontinuity, and anisotropy. Natural flaws, e.g., joints, weak planes, faults, cleavage, schistory, etc., lead to non-linear geomechanical behaviors in naturally fractured rock mass [1; 2; 3; 4; 5]. The long-term stability of rock mass is obviously influenced by these discontinuities; notably, the structural deterioration of rock mass is accelerated when exposed to cyclic or fatigue loading [6; 7; 8; 9]. Therefore, investigations into the mechanical responses of flawed rock mass are crucial to maintain the operation of underground rock infrastructures.  
Laboratory mechanical tests like uniaxial compression or triaxial compression are often used to study the cyclic or fatigue mechanical properties of rock. These properties are affected not only by the geometric structure of flaws, but also by several loading parameters such as amplitude, frequency, mean stress, and the number of fatigue cycles [10; 11; 12; 13]. Fatigue loading tests are categorized into constant-amplitude and variable-amplitude tests. Previous studies have primarily explored the influence of loading parameters--such as waveform, maximum stress, mean stress, amplitude, frequency, and cycle number--on the strength and deformation characteristics of rocks. The sinusoidal signal is widely acknowledged as closely resembling the dynamic waveforms generated by blast vibrations, earthquakes, or rock bursts. Therefore, sinusoidal stress waveforms are commonly employed in fatigue mechanical tests [14; 15]. Variable-amplitude cyclic loading, involving multiple levels, is considered more representative of the actual stress conditions in engineered rock masses than constant-amplitude loading tests. Field monitoring data also confirm that rock masses experience multi-level fatigue loading [16; 17; 18]. In underground rock engineering, blasting- or mining-induced earthquakes impose non-constant-amplitude fatigue loading on the surrounding rock structures. Numerous studies have investigated intact or flawed rocks subjected to variable-amplitude or variable-frequency fatigue loading, revealing insights into rock deformation, damage propagation, and failure patterns from both mesoscopic and macroscopic perspectives [11, 19, 20]. Nevertheless, the impact of rock bridge segments on rock fatigue response remains uncertain. Non-connected rock bridge segments within the rock mass serves as high-energy storage zones that impede fissure connection and associated instability events.  
In summary, there is a paucity of studies investigating the impact of rock bridge length (RBL) on volumetric deformation, damping characteristics, and damage evolution. Therefore, this study conducted a series of increasing-amplitude cyclic loading tests on marble samples with RBLs of 10 mm, 20 mm, 30 mm, and 40 mm. Fatigue mechanical properties, including damping effects, fatigue deformation, strain incremental rate, damage evolution, and fracture development, were systematically analyzed. The findings enhance the understanding of how rock bridge segments influence the fracture evolution of flawed rock and offer new insights into their role in resisting external fatigue disturbances.