## 1 Introduction  
It is well-established that weight reduction and increases in efficiency are the design requirements of next-generation aero-engines. Without changing the structural layout, new types of materials with unique properties should be fabricated [1; 2; 3]. From this perspective, metal matrix composites have a higher specific strength and specific stiffness than conventional metallic materials [4; 5]. The SiC\({}_{\text{f}}\)/TC\({}_{\text{4}}\) composite is a type of long fibre-reinforced metal matrix composite that has been widely used in the shafts of many engines to achieve the above-mentioned objectives [6; 7]. The damage processes of SiC\({}_{\text{f}}\)/TC\({}_{\text{4}}\) composites, which are regarded to be anisotropic materials, are complex and diverse. Further, the shaft also needs to withstand torque as it is a relatively important transmission component [8]. Therefore, a deep understanding of the damage evolution and failure modes of composite shafts can provide guidance for their application in aero-engines. Shafts are core components in all mechanical equipment because they support rotating devices and transmit motion, torque, or bending moment. A large number of experimental studies and simulation calculations have been performed on the damage mechanism, failure mode, and fatigue fracture of these components [9; 10]. GE and Pratt & Whitney [11]successfully applied a SiC\({}_{\text{f}}\)/TC\({}_{4}\) composite turbofan shaft to an XTE-45 engine, with a total length of 127 cm and a layup of [\(\pm\)15\({}^{\circ}\)]\({}_{\text{36}}\). GE Aero Engines developed four specimens of fibre-reinforced titanium matrix composite cylindrical shafts. They conducted vibration modal analysis and failure mode analysis experiments under normal- and high-temperature conditions [12; 13]. Further, [PERSON] analysed the failure causes of a combat jet aircraft rudder shaft (RS) during flight missions using a finite element method, proposing new design suggestions [14]. In 2005, the University of Nottingham and the European Commission jointly conducted the VITAL project to develop a total of 24 specimens with SiC\({}_{\text{f}}\)/Ti-6 Al-4V composite shafts. Moreover, stiffness tests, fatigue tests, and failure mode analyses were carried out, as well as numerical calculations and model validation [15; 16]. All studies on shaft damage, failure, and fatigue aimed to better transfer the load to mechanical equipment and yield a more optimal design of the shaft [17; 18]. At present, there are various methods for progressive damage analysis, most of which focus on numerical calculations and experimental studies [19; 20]. Usually, a numerical model is established, and an appropriate failure criterion is selected according to the computation module [21]. For example, [PERSON] [22] studied the shear behaviour of a stiffened composite panel with a notch through numerical simulation using a three-dimensional progressive degradation model, establishing a new method to take into account the gradual intra-laminar degradation of composite laminates. [PERSON] et al. [23] also used a single-cell model with fibre hexagonal distribution to predict the fibre's bundle properties and then simulated the progressive damage process of triaxial woven composites. In another study, [PERSON] [24] established a new method to model the interactions between interface debonding and skin damage, based on the cohesive zone model and continuum damage mechanics scheme; this method can facilitate progressive damage analysis of a blade under extreme loading. [PERSON] used a novel method to evaluate fatigue damage under sub-synchronous oscillation incidents, which utilizes field data, modal decomposition, and superposition, combining structural full-size finite-element modal analysis with the refined finite-element structural analysis of specific parts [25]. [PERSON] et al.