ways movement determined by graphical analysis at time 0.24 s. The value is somewhat approximate, but the corresponding values in the force change little around this time in the data set. The compression force calculated from the finite element model (Abaqus) showed no significant change in value at the onset of sideways movement as can be seen in Fig. 9 for the exact same model as shown in Fig. 8. The strain energy from the finite element model was also analyzed and did not reveal a sign of sudden change at the onset of sideways movement of RP-2. Thus, there is no observed significant change in strain energy that indicates the onset of the buckling phenomenon observed.
Compression tests were done with a longitudinal extensometer and no transversal extensometer. The exact position of the thinnest section is difficult to locate in an hourglass-shaped specimen. Even a small misposition will influence the results as it is difficult to place a transversal mechanical extensometer on the exact place where the diameter is measured.
The experiments were designed to identify a critical force level on which the sideways movement of the specimens started. Therefore, the distance between the grips was set to 100 mm as can be seen from Fig. 6. This distance was marked with a marker on the specimens and introduced a small uncertainty for the experiment. A constant distance between the grips makes the experiments reproducible. Two specimens for each of the geometries described in Table 2 were tested.
The longitudinal extensometer was positioned in the parallel section outside the hourglass-shaped area. The measurement length of the extensometer was 50 mm. This makes the measurements from the tests reproducible as the diameter is constant at the two mounting points and the position of the mounting does not change as it is outside the area of plastic deformation. The extensometer measures both compression and elongation within the range of the instrument. The extensometer in use is documented with a linearity of 0.15% of full scale from the supplier Instron 2015 (reference manual). The setup is shown in Fig. 10.
The compression tests were force controlled. The tests had a power increase of 500 N/s up to the set maximum limit, which depended on the specimen geometry, and the critical buckling force calculated from the finite element model (Abaqus).
In order to identify the critical force for sideways movement, the experimental data need to be analyzed. The sideways movement is initiated by exceeding a critical force. In order to identify the onset of buckling, change in strain to force was analyzed by use of the following equation related to flexibility. Flexibility is derivative of displacement with respect to force, and this equation is correlated to flexibility by a constant factor since displacement is replaced by strain.
The number of datapoints for each specimen is typically above 4500, so the change from point to point is small. Therefore, the derivative gives a good indicator for changes both the force and the strain. The absolute value gives a more readable figure. A typical example of the critical force is given in Fig. 11. The geometry of this specimen can be seen in Table 2, with a smaller diameter of 7 mm and a radius of curvature of 56 mm.
The recorded data are somewhat noisy and no data filters are used. Here, the derivative shows the first clear peak which is interpreted as the critical force for onset of sideways movement. A closer look at the recorded data shows that the maximum peak of the derivative is caused by a change in the measured longitudinal strain not followed by a corresponding change in force. This explains how the sideways movement appears. The derivative of the strain to force curve in Fig. 11 shows a clear peak at 22600 N. The corresponding force-displacement curve in Fig. 12 does not show any clear sign of onset of buckling.
Results and discussion
The model showed a reasonable correlation with the experimental results, as shown in Table 3. The deviation between the model and the experiments is maximum 11