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. 6. Cycles 4 lc - 9 lc . The slopes of the upper E x ↑ and lower E x ↓ branches

Fig. 7. Cycles 4 lc - 8 lc . The distances between the slopes of the branches

Double differentiation of the equations of the upper and lower branches of the hysteresis loops makes it possible to obtain the equations for the rates of change in slopes of the upper dE x /dε x ↑ and lower dE x /dε x ↓ branches (see Fig. 8), as well as equations for the distance between these two rates, dE x /dε x ↑ - dE x /dε x ↓ (see Fig. 9). For each hysteresis loop: 1. the rate of change in the slope of the lower branch is positive everywhere and, starting from the 6 lc cycle, everywhere is greater than the rate of change in the slope of the upper branch; 2. near the left (except for the 8 lc cycle) and right vertices, the rate of change in the slope of the upper branch passes into the area of negative values, which indicates a relative decrease in the stiffness of the material in these sections of the stress-strain curve; 3. the points of intersection of the dE x /de x ↑ curve with the x-axis determine the location of the inflection points of the upper branch; 4. the lower branch does not have inflection points.

Fig. 8. Cycles 4 lc - 9 lc . The rates of change in the slopes of the upper dE x /d x ↑ and lower dE x /d x ↓ branches

Fig. 9. Cycles 4 lc - 8 lc . The distances between the rates of change of the slopes of the branches

The graphs of the polynomial equations of the upper and lower branches of the hysteresis loops in Fig. 3 are approximately linear, but not exactly so. Each of the seventeen branch equations can be expressed as the sum of linear and non-linear components: σ x ↑ = σ xl ↑ + σ xnl ↑ , σ x ↓ = σ xl ↓ + σ xnl ↓ . The linear components are obtained as a result of approximating the experimental data for simultaneously measured ε x and σ x in sections of increasing and decreasing displacements using polynomials of first degree. We obtain the non-linear components by subtracting the linear components of the branches from the equations of the branches. The linear components of the upper σ xl ↑ and lower σ xl ↓ branches of the hysteresis loops, as well as the distances σ xl ↑ - σ xl ↓ between them, are shown in Fig. 10 and 11. The linear components of the upper branches are positioned above the linear components of the lower branches.