PSI - Issue 65

P.B. Severov / Procedia Structural Integrity 65 (2024) 215–224 P.B. Severov / Structural Integrity Procedia 00 (2024) 000–000

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Fig. 14. Cycles 1 lc - 9 lc . Non-linear components of the upper σ xnl ↑ and lower σ xnl ↓ branches

Fig. 15. Cycles 1 lc - 8 lc . The distances between the non-linear components of the branches

Fig. 16 and 17 show the slopes of the non-linear components of the upper E xnl ↑ and lower E xnl ↓ branches and the distances E xnl ↑ - E xnl ↓ between them. The slope graphs of the non-linear components of the branches in Fig. 16 can also be obtained by shifting the slope graphs of the branches in Fig. 6 in the negative direction of the y-axis by the value equal to the slope of the linear component of each branch, as shown in Fig. 12. The graphs of the distances between the slopes of the non-linear components of the branches in Fig. 17 are shifted relative to the graphs of the distances between the slopes of the branches in Fig. 7 by the value equal to the difference in the slopes of the E xnl ↓ - E xnl ↑ linear components of the lower and upper branches. This shift occurs in a positive direction along the y-axis. The difference in the slopes of the linear components of the branches (Fig. 12) varies slightly from cycle to cycle. At the left vertices of the hysteresis loops, the slopes of the non-linear components of the upper branches are greater than the slopes of the non-linear components of the lower branches. At the right vertices, the slopes of the non-linear components of the lower branches are significantly greater. The non-linear components of the upper branches have one extremum in cycles 1 lc - 6 lc , and two in cycles 7 lc - 9 lc . The non-linear components of the lower branches in cycles 1 lc - 8 lc have strictly one extremum.

Fig. 16. Cycles 4 lc - 9 lc . The slopes of the non-linear components of the upper E xnl ↑ and lower E xnl ↓ branches

Fig. 17. Cycles 4lc - 8lc. The distances between the slopes of the non-linear components of the branches

Since the rates of change in the slopes of the linear components of the branches are zero, the rates of change in the slopes of the non-linear components of the branches are equal, respectively, to the rates of change in the slopes of the branches: dE xnl /dε x ↑ = dE x /dε x ↑ и dE xnl /dε x ↓ = dE x /dε x ↓ (Fig. 8). The rates of change in the slopes of the non linear components of the lower branches are almost everywhere greater than the rates of change in the slopes of the non-linear components of the upper branches. This is especially clearly observed near the vertices of the hysteresis loops. The non-linear components of the upper branches have two inflection points in cycles 4 lc - 7 lc , 9 lc , and one in cycle 8 lc . The non-linear components of the lower branches in cycles 4 lc - 8 lc do not have inflection points. The