Improved model

It was further attempted to find a modification of the numerical model that would give results closer to the experiments in the shock tube. Five alternative models were proposed and are listed in Table 8.14. The parameters in the proposed models were gradually changed compared to the initial model, with the last model being the most modified. It was expected that the first model would provide the largest capacity. In the experiments, the concrete plates established though-thickness cracks for a load of approximately 18.5 bar and a total collapse for a load of approximately 29.5 bar. It should be emphasised that the load of 18.5 bar corresponded to a Friedlander equation with an abrupt rise to peak pressure, while the load of 29.5 bar happened more gradually.

Table 8.14: Alternative models.

Parameters Model 1 Model 2 Model 3 Model 4 Model 5

A 0.79 0.79 0.79 0.5 0.5

B 1.6 1.2 1.2 1.0 1.0

n 0.69 0.9 0.9 1.0 1.0

e^f_min 0.005 0.004 0.004 0.004 0.002

D₁ 0.02 0.01 0.01 0.01 0.01

f_c [MPa] 55.21 55.21 55.21 55.21 55.21

f_ct [MPa] 3.80 3.80 3.80 3.80 3.80

Erode command 2 2 0 2 0

# Elements 8 8 8 10 10

Boundary conditions Plates Plates Plates Plates Plates

Based on the material parameter study, all the proposed models used the cylinder strength instead of the cube strength. The tensile strength had little impact in the material parameter study, but to stay conservative, the lowest value found through material tests was used in all models. The boundary conditions would be modelled as rigid plates for all models as this was assumed to be the best physical approach even though the computational cost would

CHAPTER 8. NUMERICAL STUDIES

increase. To reduce the strength of all the models, the damage parameters were set to lower values. This was done to achieve a more brittle model and a larger damage accumulation.

Model 1

In the first model, all parameters except for the strength parametersA,B andnwere changed in order to reduce the strength. Table 8.15 presents the collapse load for Model 1. The load provided an upper limit of 5 bar and the exact collapse load would lie somewhere between 75 and 80 bar. This also applies to the rest of the tables presented for the alternative models.

Table 8.15: Results for Model 1.

Model Collapse load Difference from initial model

[MPa] [%]

1 80 -42.9

As shown in Table 8.15, the model managed to reduce the strength of the initial model with approximately 42.9%. This was still far too strong compared with the physical experiments.

Model 2

The damage parameters in the second model were reduced to the lowest value investigated in the parameter study. This was done to speed up the damage accumulation and to reduce the strength further. The fracture strength parameters B and n were also changed, but not to the extremes in their investigated bounds. This was done so that the input values would not differ too much from the values found in the literature.

Table 8.16: Results for Model 2.

Model Collapse load Difference from initial model

[MPa] [%]

2 52.5 -63.2

CHAPTER 8. NUMERICAL STUDIES Table 8.16 shows that the strength was close to the peak reflected pressure found in the Eulerian simulation of the experiment on the fourth plate (see Figure 6.9 in Section 6.2.3).

Since the numerical simulation was performed without any FSI, the reflected load on the plate would be greater than in the actual experiment.

Model 3

The only difference between Model 2 and Model 3 was the erosion criteria. In the parameter study, it was observed that the response from the two approaches deviated (see Section 8.2.3).

However, a coarser mesh was used, and it would therefore be interesting to see if this effect vanished for a more refined mesh.

Table 8.17: Results for Model 3.

Model Collapse load Difference from initial model

[MPa] [%]

3 50 -64.9

(a) Model 2 (t=1.0 ms) (b) Model 3 (t=1.0 ms) Figure 8.23: Crack patterns with different treatment of failure.

CHAPTER 8. NUMERICAL STUDIES

Table 8.17 shows that the reduction in peak pressure was small compared to Model 2. The damage pattern and the failure mechanism was similar, which indicated that the difference between using no erosion and node splitting was small when the mesh was sufficiently refined (see Figure 8.23).

Model 4

In order to develop a model that would collapse for a peak pressure seen in the experiments, further changes were made to the model. Node splitting was selected and the mesh was increased to 10 elements over the thickness. The cohesive strength A, was reduced to its minimum investigated value, and the fracture parameters B and n were also pushed to the value that would give the most reduction in peak pressure. This was done to produce the weakest model possible within the selected parameter bounds.

Table 8.18: Results for Model 4.

Model Collapse load Difference from initial model

[MPa] [%]

4 40 -71.9

Table 8.18 shows that Model 4 also collapsed for a higher load than the load of 29.5 bar found in the experiments. However, the model collapsed for a pressure between the peak pressure found in the Europlexus simulation with a rigid plate, and with the experiment.

Model 5

A fifth and last model was developed to see if it was possible to get the model to collapse for an even lower pressure. This was done by reducing e^f_min to half of the value Polanco et al.

[41] had suggested in their model.

CHAPTER 8. NUMERICAL STUDIES Table 8.19: Results for Model 5.

Model Collapse load Difference from initial model

[MPa] [%]

5 32.5 -77.2

Model 5 gave the lowest capacity of all the proposed models, and the collapse load was close to the pressure of 29.5 bar registered by sensor 409 in the fourth experiment. The model was therefore chosen to recreate the experiments performed in the shock tube.