PSI - Issue 65

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

218 4

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

Fig. 3. Cycles 1 lc - 9 lc . The upper σ x ↑ and lower σ x ↓ branches of the hysteresis loops

Fig. 4 and 5 for each hysteresis loop show the distance between the branches along the y-axis and the dissipation of mechanical energy per unit volume of material during repeated loading. The hysteresis loops at the left and right vertices are open due to the removal of the s max = const and s min = const sections from consideration. For each loop, the domain of definition of the upper branch (ε xmin ↑ , ε xmax ↑ ) does not coincide with the domain of definition of the lower branch (ε xmin ↓ , ε xmax ↓ ). The upper branch is shifted along the lower branch in the direction from the origin of the coordinate axes. The domain of definition for the hysteresis loop width is set as (ε xmin ↑ , ε xmax ↓ ). The dissipation of mechanical energy per unit volume of material in hysteresis loops can be obtained by integrating the equations, whose graphs are shown in Fig. 4. The integration is done over the range of ε from ε xmin to ε x . All curves in Fig. 5 have inflection points, whose abscissas correspond to the abscissas of the maxima of the curves in Fig. 4. The width of the hysteresis loop can be interpreted as the rate of mechanical energy dissipation per unit volume of material, as a function of ε x . The maximum rate of mechanical energy dissipation in the hysteresis loop is equal to the largest width of the hysteresis loop.

Fig. 4. Cycles 1 lc - 8 lc . The distance between the branches or the width of the hysteresis loops

Fig. 5. Cycles 1 lc - 8 lc . The dissipation of mechanical energy per unit volume of material in hysteresis loops

The differentiation of the equations of the upper and lower branches of the hysteresis loops, the graphs of which are shown in Fig. 3, makes it possible to obtain the equations for the slopes of the upper E x ↑ and lower E x ↓ branches as a function of ε x (Fig. 6), as well as the equations for the distance E x ↑ - E x ↓ between them (Fig. 7). For each hysteresis loop: 1. the slopes of the upper and lower branches are equal at the point e x of the largest value of the width of the hysteresis loop; 2. before the specified point, the slope of the upper branch is greater than the slope of the lower branch, after the specified point, the slope of the lower branch is greater than the slope of the upper branch; 3. positive values of the slopes of the upper and lower branches indicate an increase in the upper and lower branches; 4. the extremes on the slope curve of the upper branch determine the location of the inflection points of the upper branch graph near the left and right vertices of the loop; 5. the absence of extremes on the slope curve of the lower branch means that there are no inflection points of the graph of the lower branch.