###### Abstract  
Although the material has developed micro-scale cracks, the micro-cracks stop propagating and transform into non-propagating cracks (NPCs) under fatigue limit loading. The movement of the crack tip position caused by the non-propagating crack will generate a small change in the notch geometry, which easily triggers the geometric size effect. Since the critical distance method focuses on evaluating the limit of fatigue in terms of the material's cracking conditions, the present study attempted to develop a notched fatigue analysis model to consider the effect of non-propagating cracks on fatigue limit prediction in the critical distance method framework. The effectiveness and capability of the proposed model were demonstrated by the fatigue experimental data of Q345 qD low carbon steel.  
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Footnote †: journal: Applied sciences  
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Footnote 1: Department of Civil Engineering, The Changsha University of Science &Technology, Changsha 410114, China  
for the effect of notch size. [PERSON] et al. [24] pointed out that the critical dimension effect of the crankshaft can be considered by superimposing the stress gradient over the conventional critical distance.  
Numerous evidence [25; 26; 27; 28; 29; 30; 31; 32; 33] has revealed that although components may already have generated microcracks, the microcracks stopped propagating and transformed into non-propagating cracks (NPC) under fatigue limit load. In many engineering applications, the crack initiation stage does not exist, and some initial microcracks generated by the manufacturing or processing process pre-exist in the engineering component. Therefore, the fatigue limit of materials in practical structural engineering, such as steel [33; 34], is mainly determined by non-propagating fatigue crack behavior. Different studies have shown that the fatigue limit of materials should be evaluated based on the threshold stress of microscopic non-propagating crack growth [33; 35]. In view of this, numerous models based on the NPC concept have been established for fatigue analysis. Among these, [PERSON] et al. [36] proposed a fracture mechanics model for evaluating fretting fatigue strength with non-propagating cracks. [PERSON] et al. [37] used the non-propagating crack criterion to estimate the durability limit of welded joints. [PERSON] et al. [38] provided a model that combines undercut depth, maximum non-propagating crack length and threshold stress value to determine the minimum durability limit.  
Although [PERSON] and [PERSON] later illustrated that the fatigue limit can be defined in terms of the formation of non-propagating cracks emanating from components' weakest points [39], they did not consider the movement of the crack tip position caused by the non-propagating crack and its effect. In addition, the movement of the crack position changes the original geometry of the notch, which easily triggers the geometric size effect. Therefore, the present study attempted to develop a notched fatigue analysis model to consider the effect of non-propagating cracks on fatigue limit prediction in the critical distance method framework. The paper is organized as follows: in Section 2, the details and results of fatigue test for Q345 qD low carbon steel are described; in Section 3, the critical distance theory is briefly reviewed, and the fatigue limit of Q345 qD low carbon steel is predicted by the critical distance method; in Section 4, the critical distance model recommended in this study is proposed, and related analysis and discussion are carried out; finally, some conclusions are drawn from the current study in Section 5.