3.2.8 Simple proportional and nonproportional cyclic tests

Product: ABAQUS/Standard  

This example illustrates the process of calibrating the nonlinear isotropic/kinematic hardening model using test data from a uniaxial, symmetric strain-controlled, cyclic experiment. It also illustrates the limitations of the model under multiaxial loading conditions when the material properties are calibrated with uniaxial test data.

Three different simulations are performed in this example. The simulations include a uniaxial, symmetric strain-controlled experiment; a uniaxial, unsymmetric strain-controlled experiment; and a multiaxial tension-torsion experiment. The model predictions are compared with experimental test data for OFHC copper (Anand, 1996). The simulations show that the model captures the response of the material accurately when the experiment that is used to calibrate the model is simulated. However, it only approximates the behavior of the material when the loading does not correspond to the loading of the calibration experiment.

Models for metals subjected to cyclic loading, Section 18.2.2 of the ABAQUS Analysis User's Manual, contains a description of the model and its use; and a mathematical description of the model is presented in Models for metals subjected to cyclic loading, Section 4.3.5 of the ABAQUS Theory Manual.

Problem description

Calibration of the model

The model is calibrated using test data from a uniaxial experiment (Figure 3.2.8–1) obtained at a strain range  1.5%. Both the kinematic component and isotropic hardening component of the model are calibrated.

The shape of the first cycle differs from the shape of subsequent cycles, suggesting that the kinematic hardening component is a function of the cycle number. Since the model does not allow for such a dependency, a representative shape must be chosen. The objective in this example is to compare the model predictions with test data over many cycles. The stabilized cycle is, therefore, chosen for calibration. If the model were being used to simulate only one or two load cycles, it would be more appropriate to use the first loading cycle for calibration.

The second half of the saturated cycle used for calibrating the kinematic hardening material parameters C and is shown in Figure 3.2.8–2. The data are entered as values of yield stress, , versus plastic strain, , on the data lines of the *PLASTIC, HARDENING=COMBINED, DATA TYPE=STABILIZED option, where

with the total strain for data point i, and The onset of yield is taken as  46.9 MPa. The calibration yields  33.55 GPa and  701.3; these quantities are reported in the results file.

The isotropic hardening component is calibrated next. Isotropic hardening defines the evolution of the elastic range as a function of equivalent plastic strain. The size of the elastic range can be determined easily at points where the loading is reversed as half the difference between the yield stress in tension and compression. For the stabilized cycle the size of the elastic range is 96.2 MPa. The corresponding values of equivalent plastic strain are obtained by assuming that the test is approximately performed as a symmetric plastic strain-controlled experiment, where

and is an averaged yield stress over all the cycles. is taken as 75.0 MPa for this material. With this assumption the equivalent plastic strain is obtained as

where i is the cycle number. This approximation yields a value of  25.16% for the last cycle ( 10). The resulting data are entered in tabulated form on the data lines of the *CYCLIC HARDENING option. The change in elastic range during the first half-cycle is specified as zero to compensate for the difference in the shape of this cycle compared to subsequent cycles.

Results and discussion

Acknowledgment

ABAQUS would like to thank Professor L. Anand of the Massachusetts Institute of Technology for providing the experimental test data.

Input files

Reference

Figures

Figure 3.2.8–1 Symmetric strain cyclic test data.

Figure 3.2.8–2 The last half cycle of test data is used to calibrate the kinematic hardening component.

Figure 3.2.8–3 Comparison of the calibrated model and the test data for the symmetric strain cycle experiment.

Figure 3.2.8–4 Comparison of the calibrated model and the test data for the unsymmetric strain cycle experiment.

Figure 3.2.8–5 Comparison of the calibrated model and the test data for the tension-torsion cycle experiment.