The push–pull fatigue characteristics of the peak-aged Mg–0.2Zn–0.5Zr alloys with different addition levels of neodymium (Nd) have been investigated. The fatigue strength (σf) of the Mg–xNd–0.2Zn–0.5Zr (NZx0K) alloy increases proportionally with the increase of the Nd content (C Nd) as follows: σf (T6) ≈ (13.8–14.0) C Nd + 46 (for x between 0 and 3.0 wt%). The cyclic stress amplitude also increases but the plastic strain value decreases with the increase of the Nd content. The studied alloys exhibit the strain hardening followed by cyclic softening during fatigue test. During the low-cycle fatigue (LCF) test, the cracks originate from the cyclic deformation and cumulative damage. In high-cycle fatigue (HCF), the failure is due to the cyclic deformation and damage irreversibly caused by environment-assisted cyclic slip. The LCF lives of the alloys fitted well with the Coffin–Manson relation and Basquin laws, the three-parameter equation, and the energy-based concepts. The developed multi-scale fatigue (MSF) life models can be used to predict the LCF and HCF lives of the alloys. Among these models, the MSF life can well capture the influence of Nd addition on fatigue.

A series of characterization tests were performed to elucidate the high-cycle fatigue (HCF) behavior in SAE 1050 steel subjected to tensile elastic pre-deformation at different strain rates. In the pre-strained stage, the deformation was maintained constant at 0.16%, which was close to the low yield point at strain rates ranging from 10−5 s−1 to 10−2 s−1. Although pre-deformation occurred entirely in the elastic regime, using different pre-straining rates resulted in the occurrence of heterogeneous microscopic strain at different sites and locations during subsequent fatigue tests. It was found that the effect of pre-straining rate on crack initiation and crack propagation was not monotonous and was influenced by the homogeneity of deformation within grain boundaries, the integrity of the boundary structure, and the fracture toughness. In addition, the rough set theory model was introduced for the attribute reduction of characteristic parameters and provided a scientific basis to establish the fatigue model. The model was able to effectively predict the lifetime of the process of HCF in pre-strained steel. Hence, the pre-straining rate should be an important boundary condition in further studies.