PSI - Issue 42

6

Monika Středulová et al. / Procedia Structural Integrity 42 (2022) 1537– 1544 M. Strˇedulova´ et al. / Structural Integrity Procedia 00 (2019) 000–000

1542

2 . 2

free simulations fixed simulations experiments

2 . 0

1 . 8

1 . 4 P / P R2 . 5 1 . 6

1 . 2

1 . 0

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1 . 5

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slenderness ratio

Fig. 4. Comparison of peak forces obtained on specimens of varying slenderness ratios. Experimental results (Lisztwan et al., 2021), friction free and full friction simulations with free or completely restricted lateral movement, respectively.

as observed in the experimental setting and described in literature. Peak force P obtained by the calculation and normalized with respect to the peak force of slenderness ratio 2.5, P R 2 . 5 , has been used for the purpose of comparison. Fig. 4 shows the resulting average values along with plus and minus standard deviations. The data show a clear influence of the lateral movement restriction on the bases of the cylinder. Slenderness ratio of 2 gives only slightly di ff erent peak force to the highest slenderness ratio, all within the bounds of standard deviation. Simulations where free lateral movement was allowed show only minimum di ff erence in the peak force as the slenderness ratio approaches 1, with the ratio P / P R 2 . 5 oscillating around 1. Averaged value of the P / P R 2 . 5 ratio of slenderness ratios 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 2.0 is 1.016 ± 0.02. In contrast, simulations where no lateral movement of bases particles was allowed show a significant increase in the peak load for the smallest slenderness ratios. While slenderness ratio of 1.5 shows increase of the peak load only by 6.19 %, slenderness ratio 1.1 shows increase of 35.5 %. Experimental observations provide P / P R 2 . 5 ratio in between the friction free and the full friction simulations as expected. Clearly, none of the extreme simulation cases is suitable to describe the friction in a realistic way. The lowest slenderness ratio which was tested shows increase in peak load of about 26.5 %. The ratio 1.5 shows increase of 4.7 %, which is smaller than the full friction simulations of the same slenderness ratio. The stress–strain diagrams of simulations leading to the above described results are showed in Fig. 5. All the curves correctly show identical slope in the linear part of the diagram, which is independent of the boundary conditions. Peak load di ff ers in a manner described in the previous paragraphs. The post-peak response is strongly influenced by the slenderness ratio of the specimen for both cases of boundary conditions. The slope of the curve shows higher ductility for specimens of lower slenderness ratio, while high slenderness ratio specimens exhibit rather brittle behav ior in comparison. The trend may be observed in both friction free and full friction simulations, although it is more pronounced in the latter set of results. Note that the full friction model remains in the hardening regime till the end of the simulation for slenderness ratio 1.0. Most likely the formulation of the constitutive model in the state of the triaxial compression is not ideal and needs some improvement. The same response type occurs also later for models with high fiction parameter µ . Therefore, numerical results with high friction and slenderness ration 1.0 shall be considered unreliable. In a subsequent stage, the simulations were run considering following friction coe ffi cients µ : 0.005, 0.1, 0.2, 0.4. Five realizations di ff ering in random internal meso-structure (used previously for friction free and full friction models) were also used this time. The lowest friction coe ffi cient shows similar results to those of free friction boundary conditions (Fig. 6, left-hand side). Mean value of the P / P R 2 . 5 ratio for µ = 0 . 005 shows only negligibly variations. The results also follow the pat tern showed by the friction free simulations, suggesting a strong influence of the internal meso-structure geometry and need for more realizations. However, comparison of stress–strain curves of ratio 1.0 (Fig. 6, right-hand side) shows di ff erences in the post-peak behavior; even small frictional resistance leads to an increase in ductility. Application of larger friction coe ffi cients (0.1, 0.2, 0.4) increases the rate with which the peak load grows for low slenderness ratio