PSI - Issue 68

Rita Dantas et al. / Procedia Structural Integrity 68 (2025) 901–907 Rita Dantas / Structural Integrity Procedia 00 (2024) 000–000

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structures, slip planes are usually characterized by larger atomic distances and smaller interatomic distances, which leads to higher values of Peierls-Nabarro stress, as shown in Fig. 2. As consequence, at higher frequencies, less slip planes are activated and higher resistance to plastic deformation is observed. Nonetheless, in steel alloys with higher ultimate strength, the frequency e ff ect can diminishes, and in some cases, it vanishes completely. For certain steels, the dominance of ultimate strength over lattice structure can be explained by the presence of obstacles which are not considered in Peierls-Nabarro stress. These obstacles restrain the dislocation movement, which leads to the improvement of fatigue resistance (Hong et al , 2023; Zhao et al , 2012). The frequency e ff ect in materials with hcp (hexagonal closed pack) crystal structures should be similar to bcc crystals, considering they have even fewer slip planes. However, alloys of hcp structure are usually characterized by high ultimate strength, which promotes less sensitivity to frequency e ff ect (Hong et al , 2023; Morrissey et al , 1999). In addition, the strain rate e ff ect can also be explained based on the relation between the shear stress require to move a dislocation and the velocity of this movement (Johnston and Gilman , 1960; Stein and Low , 1960; Zhao et al , 2012). Stein and Low (1960) analysed, experimentally, the relation between shear stress applied and velocity of a dislocation in silicon crystals, and defined the following equation: V = V 0 τ τ 0 m (2) where V is the dislocation velocity, m is a material constant, τ is the shear stress applied into a slip plane and τ 0 is the shear stress required to achieve a dislocation velocity V 0 = 1 cm / s . From this equation can be concluded that the shear stress increases with the velocity of movement as shown in the plot of Fig. 3 (b), where are compared di ff erent ratios between them. The establishment of Eq. 2 evidences the dependency of the dislocation movement on the velocity and, conse quently, on the frequency of loading. Furthermore, similar to Eq. 1, this relation varies with the material characteris tics, since the shear stress required to move a dislocation can be more or less a ff ected by the increase of velocity. However, to quantify the dependency of fatigue damage on the velocity of loading, the distance travelled by a dislocation ( L ) can be defined based on Eq. 2. Firstly, the velocity of a dislocation can be defined as derivative of the distance travelled in order to time ( t ), as follows: Consequently, the distance a dislocation moves in a quarter of cycle ( T / 4) is defined as (Zhao et al , 2012): L = T / 4 0 Vdt (5) Considering the loading case of a cyclic stress with a sinusoidal shape, shear stress can be defined as: τ ( t ) = τ sin ( ω t ) = τ sin (2 π f t ) (6) where ω is the angular frequency and f is the frequency. Thus, the dislocation velocity can be described as: V = V 0 τ sin (2 π f t ) τ 0 m (7) By combining Eq. 5 with Eq. 7, the distance travelled by a dislocation can be defined as: L = T / 4 0 V 0 τ sin (2 π f t ) τ 0 m dt (8) V = dL dt (3) then, the equation can be rewritten and integrated: dL = Vdt (4)