hange in strain and/or force. To predict the critical buckling force, finite element models were made using the finite element code (Abaqus Dassault Systèmes®). The material model in the commercially available code (Abaqus) was calibrated based on tensile tests of the material. The model was validated by experimental tests in order to verify the critical buckling force for the actual material.
In this work, the specimens were tested in an Instron servo-hydraulic testing machine (Instron 8800). The specific machine is equipped with a climate chamber for testing temperatures ranging from − 70 to 350 °C and therefore has elongated rods for the allowance of the specimen mounting heads inside the chamber. This gives a wide experimental range. The elongation rods are treaded onto the piston rod of the machine. Thereby, some mechanical flexibility is introduced. However, the experiments must be designed so that buckling is avoided during the compression phase of the cycle.
Methods and experimental design
Material, constitutive model, and parameter identification
The material used for these tests is a commercial quality hot rolled structural steel S355J2-1.0577. The specimens were machined from a Ø = 16-mm bar which were approved according to EN10025-1 (2004). The minimum required yield strength is 355 MPa and the ultimate tensile strength is within 510-680 MPa according to EN10025-1 (2004). A computer numerical controlled (CNC) lathe was used to machine the specimens. A high-quality surface finish is not considered to be critical for the onset of buckling in a one-cycle compression test.
The material model is based on von Mises yield criterion, the associated flow rule, and a modified Voce law. Calibration was based on tensile tests. Design of the tensile specimens was done according to Fig. 1. Diameters were measured on the individual specimens tested. Three specimens in parallel were tested. The deformation of the specimens was measured by a longitudinal extensometer with a gage length of 12.5 mm. The tensile experiments 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 fa