## 1 Introduction  
Driven by the increasing demand on the higher mechanical performance of engineering structures, high-strength steels are steadily developed and widely applied in recent decades in multiple areas, e.g., automotive, high-speed trains, and aerospace. The high strength is mainly obtained by the in-depth and complicated design of the microstructure of steels, which also leads to new damage mechanisms for multiphase steels [1; 2; 3; 4; 5]. In terms of fatigue properties, a fatigue limit is normally evident for the conventional steels developed some decades ago [6; 7; 8]. [PERSON] et al. [9] reported that the fatigue limit increases with the increase of steel hardness and decreases with the increase of the defect area. Therefore, increasing the strength of materials is leading to a higher hardness and eventually a higher fatigue strength limit assuming the same defect size. However, for the recently developed high-strength steels, it is found that even when stress amplitude is lower than fatigue limit, the fatigue crack still will occur when fatigue life reaches a certain level [9; 10; 11; 12]. For higher stress and lower fatigue life, fatigue crack usually occurs from the component surface; for lower stress and longer fatigue life, fatigue crack usually occurs from inner microstructure [13]. The deformation heterogeneity in microstructure level, which is responsible for the damage and failure in general for steels, can be generally caused by two different mechanisms: the matrix inhomogeneity and the matrix-inclusion inhomogeneity [14; 15]. Many researchers have focused on the effect of matrix inhomogeneity on mechanical properties, their damage mechanism with experiments and modelling. [PERSON] et al. investigated the damage mechanism of a DP600 steel under different stress states experimentally [16] and via micromechanical simulation [17] and found that damage is initiated at the boundary of ferrite and martensite due to their strong deformation incompatibility. It was further revealed that the damage initiation is by nature stress state dependent due to the strain localization dependency of stress states. [PERSON] et al. [18] studied the fatigue crack initiation process of duplex stainless steel and quantified the interactions of cracks with the first microstructural barrier. [PERSON] et al. [19] investigated the stress state of martensite and ferrite before cracking of dual phase DP1000 steel and estimated for crack initiation in the martensite. [PERSON] et al. [20] reported that for a ferritic-pearlitic steel, the initiation of the cleavage fracture is located in the pearlitic phase where the ferrite and the cementite show a lamella structure with different properties. In addition to steel matrix inhomogeneity, inclusions also play a significant role during the failure process. [PERSON] et al. [21] reported the influence of inclusions on cleavage initiation and the relation between inclusion crack and steel matrix crack. [PERSON] et al. [22] studied 42 CrMo steel with different amounts of inclusions and found that the fatigue property was better with fewer inclusions. [PERSON] et al. [23] found that internal inclusions could lead to the formation of the penny-shape fine granular layer around inclusions during long-term cyclic loadings, which would cause small cracks inside the steel matrix. While investigating inclusions in steel, it is also found that there are multiple types of inclusions in steels [24; 25] generated in different processes during the steel production [26; 27]. For example, the Al\({}_{2}\)O\({}_{3}\) inclusion is produced during melting of steel while the TiN inclusions are produced during the casting of steels. Besides, the shape, size, and composition of inclusions, affected by the production method, also differ, leading to different effects on the mechanical properties of steels [28; 29]. The production processes additionally induce another level of inhomogeneity, residual stresses, based on the microstructure-level inhomogeneity.