nts were displacement controlled. A rate of 1.56 mm/min were used and a sampling frequency of 20 Hz. The strain was logged from machine displacement in addition to the longitudinal extensometer.
The variation between the stress-strain curves of the specimens tested was minimal. Results from one of the tensile tests are described in Fig. 2. The true stress-true strain and engineering stress -engineering strain curves are shown.
In order to include the yielding plateau, Voce law had to be modified by a constant to include the strain at the end of the yield plateau ε0.
where the parameters:
σys is yield stress measured from the test.
ε0 is plastic strain at the end of the yielding plateau.
Qi and Ci are constants calculated from experimental data by applying least square method.
By applying n = 2, the Voce law fitted the experimental data well. This can be seen in Fig. 3. The parameters of the Voce rule were calibrated by minimizing the following error (least square fit) by using the problem solver in spreadsheet (Excel):
where σi is the measured true stress at data point i.
σ* (εpi) is the calculated stress at the plastic strain at data point i.
k is the total number of data points used for the calibration.
The values obtained from the calibration are described in Table 1.
One can see from Fig. 3 that the Voce law model extrapolated further than the true stress-strain curve. This is because the true stress-true strain curve is only valid onto the onset of diffuse necking in elongation.
The tests were designed according to ASTM E606/606M (2012). This standard describes various geometries allowed for strain-controlled fatigue testing. This is used also for low-cycle fatigue testing. Servo-hydraulic machines are widely used for fatigue testing of steel and other high strength materials. The machine with a specimen mounted is shown in Fig. 4. For low-cycle fatigue testing the uniform specimen is not suited due to a substantial lower buckling force. Hourglass-shaped specimens are therefore better suited for low-cycle fatigue testing. The geometries tested varied with R/d ratio between the diameter and the radius of curvature as described in Fig. 5 for a general setup. The grips were circular without treads. Compression was done with specimens according to Figs. 5 and 6. Table 2 describes the different diameters and ratios used in the tests. The overall length of the specimens was larger than the prescribed values of 20 days + 4 days. This extra length was confined within the grips, so that we also could fit the longitudinal extensometer onto the specimen during testing.
Finite element model
The design of a finite element model is essential in order to predict the experimental behavior. Initially, a model for the whole machine and the specimen was made using a finite element code (Abaqus). This introduced several factors to the model that lead to assumptions and uncertainties. Such a model requires also a substantial amount of run-time of a strong computer. Therefore, a simplified model was developed, which is described in Fig. 7 (part a). It was chosen to use the explicit solver, which is commonly used to solve dynamic problems. However, the explicit method is also used in the literature to solve quasistatic problems similar to the one in this investigation shown by Romanova et al. (2019) and Hu et al. (1994). Full three-dimensional models must be made in order to capture the buckling behavior. Convergence tests of the 3D model were done by refining the element mesh until insignificant difference from the previous mesh size. In addition, an axisymmetric model in tension was modeled with a fine mesh and compared to the 3D model with the mesh used in the modeling of the compression tests. The time of the simulation was set to 1 s. This gave a sufficiently low speed of the deformation to ensure that the kinetic energy was insignificant compared to the total energy. The element chosen was the C3D8R element (8-node linear brick, reduced integration, hourglass control). Example of the geometry and mesh are shown in Fig. 7a and b, res