pectively. The number of elements were ranging from 646 to 850 elements depending on the geometry of the specimens. The mesh quality was checked with the mesh control in FEM software (Abaqus) using the default settings with respect to corner angles and aspect ratio.
The deformation was applied in the longitudinal direction to the upper part of the model, representing the upper grip. All the other degrees of freedom were fixed for this part. The displacement was given as a linear function up to a chosen maximum limit of deformation. The part within the bottom grip was modeled to be free to move in the plane normal to the longitudinal axis. All the other degrees of freedom were fixed for this part.
The boundary conditions of the presented model over-constrain the rotation of the grips and under-constrain the sideways movement of the bottom grip. Geometrical imperfections and misalignment were not included in the model.
The essential output of this model is the critical buckling force, i.e., this is the force for the onset of sideways movement. The result of the sideways movement (buckling) is shown in Fig. 7c. The simulations of the model were run on a computer equipped with a processor of type Intel® Xeon® W-2123 CPU @ 3.60 GHz and 16-GB RAM. The CPU time for the simulations was ranging from about 1400 to 2300 s. Until the point of sideways movement, the mesh quality remained good.
The finite element model (Abaqus) calculated the sideways movement of the point RP-2 (see Fig. 7a) at the bottom plane of the specimen. The onset of sideways movement was determined by taking the time-derivative of the movement in the X- and Y-direction (U1 and U3 respectively) of the point RP-2 (see Fig. 7a). A typical example is shown in Fig. 8 for R5d8. The derivative of the movements in X-Z plane is stable prior to buckling. They fluctuate around zero, but when the derivative of the movement is drifting away from zero, the sideways movement had started. This represents the onset of buckling. For example, in Fig. 8, the onset of sideways 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 sca