PSI - Issue 13

Josef Květoň et al. / Procedia Structural Integrity 13 (2018) 1367 – 1372 Josef Kveˇtonˇ / Structural Integrity Procedia 00 (2018) 000–000

1371

5

0.74 m/s

1.00 m/s

2.40 m/s

50

100

20

0

0

0

0 . 0 load [kN]

0 . 5

0

1

0

1

vertical displacement of loading point [mm]

reference sim. exp. data

1

set 1 set 2

damage

steel plate applied

0

Fig. 2. Response of the model using strain rate dependent constitutive law and steel platen at the loading point.

increase in maximum load obtained form the experimental results. Both of these phenomena can be attributed to missing strain-rate dependency in constitutive law, which should compensate for insu ffi cient resolution of the model inner structure and viscous e ff ects. The constitutive law was therefore enhanced with strain rate dependency according to Cusatis (2011) explained in sec. 2.3 Calculations using two di ff erent sets of parameters c 0 and c 1 are performed. The first one ( set 0 ) has c 0 = 1 · 10 − 5 and c 1 = 5 · 10 − 2 , and the second one ( set 1 ) has c 0 = 4 · 10 1 and c 1 = 1 · 10 3 . Response of the models is shown in Fig. 2. The reference simulation is the one from previous sec. 3.1 using m = 1. With increasing parameters value, cracking in the area of the applied load can be avoided, however, it also changes the shape of damaged zone in a way not corresponding to experimental evidence. Loading force can also be influenced by steel mass between the force sensor and concrete material, which has to be accelerated as well. In the right bottom part of Fig. 2, response of the model with additional steel piston (steel part modeled by finite elements) is shown. Value of the maximum load increases significantly, however, arbitrary loading force can be obtained using di ff erent size of this steel cylinder. Since this mass is not known to us, it is not applied for any other simulations in the presented study.

3.3. Application of random field

It has been shown that the change in material properties has large influence on model response for all the simulated displacement rates. Here, the focus is on faster loading rates from 0.74 to 2.4 m / s. For these loading velocities, 10 di ff erent random field realizations are applied. These realizations are shown in Fig.3 in the first row. Further rows show crack patterns for corresponding realization and displacement rate applied. Similarly to the change of material properties, the load-displacement response is not influenced much, but crack pattern is highly a ff ected by presence of locally stronger or weaker material.

4. Conclusion

It has been shown that present discrete particle model is able to represent the influence of loading rate on material behavior. However, it seems that some part of the rate dependency needs to be phenomenologically inserted into the constitutive relation. In case of the simulated L-shaped specimens, the peak load is highly influenced by inertia (i.e. by accelerating the mass above the loading point), while the crack pattern is highly influenced by material fracture properties. Contrary to the experimental evidence, the model predicts large zone of distributed cracking above the loading point for high loading velocities. Strain rate dependency in constitutive load helps to reduce this cracking.