PSI - Issue 65

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

220 6

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

Fig. 10. Cycles 1 lc - 9 lc . Linear components of the upper σ xl ↑ and lower σ xl ↓ branches

Fig. 11. Cycles 1 lc - 8 lc . The distance between the linear components of the branches

The slopes of the linear components of both the upper and lower branches increase from cycle to cycle (see Fig. 12). The slopes of the linear components of the upper branches (except for the first one) are less than those of the lower branches. In the first two cycles, the slopes are approximately equal in pairs. The ordinates of the intersections of the linear components of the branches with the y-axis are shown in Fig. 13. In the first two cycles, the ordinate values are also approximately equal in pairs. The close values of the slopes and ordinates of the intersection of the stress axis in loading cycles 1 and 2 give reason to assert that the linear components of the upper and lower branches in both cycle 1 and cycle 2 almost coincide.

Fig. 12. Cycles 1 lc - 9 lc . The slopes of the linear components of the upper E xl ↑ and lower E xl ↓ branches depending on the loading cycle

Fig. 13. Cycles 1 lc - 9 lc . The ordinates of the intersection points of the linear components of the upper σ xl ↑ and lower σ xl ↓ branches with the y-axis depending on the loading cycle

Fig. 14 and 15 illustrate the non-linear components of the upper and lower branches of the hysteresis loops, as well as the differences σ xnl ↑ - σ xnl ↓ between them. The non-linear components of the branches are nothing more than the deviation of the branches (Fig. 3) from their linear components (Fig. 10). The intersection points of the graphs of σ xnl ↑ and σ xnl ↓ with the x-axis in Fig. 14 determine the location of the intersection points of the upper and lower branches with their linear components. Obviously, the sum of the distances between the linear and non-linear components of the branches (see Fig. 11 and 15) is equal to the width of the hysteresis loops (Fig. 4). The non-linear components of the lower branches form a group of similar curves, each with one extremum. These curves are convex in the negative direction of the x-axis and have monotonically increasing slopes that pass through zero (see Fig. 16). The non-linear components of the upper branches change in a more complicated way. This is due to the tendency of the non-linear components of the upper branches to straighten and form a second extremum in the area of positive stress values.