PSI - Issue 31

Artyom Chirkov et al. / Procedia Structural Integrity 31 (2021) 80–85 Artyom Chirkov et al. / Structural Integrity Procedia 00 (2019) 000–000

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Tensile loading (schematically illustrated in Fig. 2a) is applied to samples with restriction of tangential sliding on loading facets: • Nodal velocities are assigned to the mesh nodes belonging to the opposite boundaries B 1 and B 2 : v z = − v , v x = 0 , v y = 0, if x i ∈ B 1 and v z = v , v x = 0 , v y = 0, if x i ∈ B 2 . • Other facets of a sample are free of stress. Since an explicit dynamic formulation of a boundary value problem (BVP) is employed, a technique of very slow load is applied in calculations in order to maintain the quasi-static conditions of loading. The whole computational time corresponds to more than 300 runs of elastic P-wave across the entire computational domain. In the next section, we provide the results of numerical modeling and discuss them.

4. Results of numerical modeling

According to the numerous experimental results, the gliding of dislocations within grains starts quite long prior to the observed upper yield point (e.g., Cottrell and Bilby (1949); Hahn (1962)). Origination of the Lu¨ders band in its turn occurs upon reaching the upper yield stress when the cross-section of the sample is the first time occupied by the plastic deformation. This process is accompanied by a drop of loading stress down to the lower yield stress in the case of ”hard” machine usage. Further propagation of the Lu¨ders band occurs at approximately constant stress level until the band (one or several) travels across the entire sample and the strain hardening stage is established.

Fig. 3. σ − ε curve obtained by modeling of sample tensile loading (a). Based on the numerical modeling data, samples demonstrate a similar behavior if the ”up-down-up” constitutive equation is employed. Due to stress concentration in the vicinity of grips (4 corners of the model simulate the zones of high-stress concentration) origination of plastic strain occurs exactly in these 4 corners prior to the observed upper yield point. Origination of two Lu¨ders bands occurs near opposite ends of samples once the upper yield point is reached. The yield plateau stage is provided by the spreading of two Lu¨ders bands towards each other. Fig. 3 illustrates an obtained σ − ε curve with model parameters listed in Table 1. Several characteristic points of the loading diagram are matched with corresponding patterns of plastic strain distribution in the computational domain. Figure 4 illustrates obtained distributions of plastic strains at instants (a)–(f). The first three instants correspond to the yield plateau stage. The front of the Lu¨ders bands is schematically highlighted by a red dashed lines in Fig. 4. To study the regularities of plastic flow in di ff erent stages, i.e. yield plateau and linear work hardening stages, we plotted the kinetic diagram of plastic flow (see Fig. 5a). Diagram illustrates the current position of single or multiple bands in the yield plateau stage and the position of maximums of plastic strain distribution along with the A-A profile (location of A-A profile is provided in Fig. 2a). Judging by the kinetic diagram, simultaneously originated