Introduction
There are many different non-linear viscoplastic material models available these days. Most of the moderns constitutive models of this type are structured in the form of parallel networks. The Bergstrom-Boyce (BB) model is my favorite two-network model of this class. This is a model that I developed while I studied at MIT many years ago. The BB-model is a built-in feature in most commercial FE codes, and works very well for predicting the large-strain response of elastomer-like materials. Note that the BB-model is part of the PRF models in Abaqus. In this article I will show that it is possible to calibrate this non-linear time-dependent model to experimental data from a single uniaxial tension test. Which is really cool, since why run more experiments than what is needed? All results shown here were generated using MCalibration.
😎 Also see our article on “Smart Testing” of polymers.
Case 1: Monotonic Tension at One Strain Rate
The plan here is to first generate stress-strain data from a known BB-model, and then attempt to calibrate a BB-model to that data using a “random” starting guess of the parameters. If the experimental data set is sufficient then the calibration should be able to recover the original material parameters that were used to create the data. If the calibration gives a different set of material parameters then the solution to the problem is not unique because not enough experimental data was provided. I will start with a single monotonic tension test case with a constant strain rate. The following figure shows the stress-strain response of the master material model.
I then perturbed the key material parameters as shown in the following image. With this modification, the material model is far from the real parameters.
The final calibration results are shown in the figure to the right. The calibrated material model fits the experimental data really well, but the final calibrated material parameters are NOT the same as the real parameters. Particularly the m parameter is too low. In other words, the BB-model cannot be calibrated to uniaxial tension at a single strain rate! That should not be surprising since the model is strain-rate dependent and we only provided data at one strain rate. More data is needed.
Case 2: Monotonic Tension of a "Smarter" Specimen
One way to provide stress-strain data at different strain rates is to use a tension specimen with a cross-sectional area that varies along the gauge section. The following image shows the procedure that I used to generate the force-displacement data that was used for the material model calibration.
I then used the “Abaqus External Solver” load case in MCalibration to calibrate the best possible BB-model to the force-displacement data. The following figure shows that the calibration is able to capture the “experimental” data, but that the calibrated parameters also in this case are not correct. In other words, the inhomogeneous strain-rate distribution in the specimen is not sufficient for calibrating the material model.
Case 3: Two Tests & Two Strain Rates
I repeated the same strategy using two strain rates (0.1/s and 0.01/s). As before, I first generated the force displacement data using the master material model. I then perturbed the material parameters and then ran the calibration. The following figure shows the final calibration results. As expected, the calibrated material parameters match the true parameters really well. Two tension tests with two strain rates is sufficient to calibrate the BB-model.
Case 4: Load-Unload at One Rate
In this example I used a single load-unload cycle at a constant strain rate. As before, I first generated the stress-strain data using the master material model. I then perturbed the material parameters and then ran the calibration. The following figure shows the final calibration results. As expected, the calibrated material parameters match the stress-strain data, but the calibrated material parameters are not correct. A single load-unload cycle with constant strain rate magnitude is NOT sufficient to calibrate the BB-model.
Case 5: Load-Relax-Load, One Rate
In this example I used a load – 30 sec relaxation – load cycle at a constant strain rate. As before, I first generated the stress-strain data using the master material model. I then perturbed the material parameters and then ran the calibration. The following figure shows the final calibration results. The calibrated material parameters match the stress-strain data, and the calibrated material parameters are correct. This type of test strategy is sufficient to calibrate the BB-model.
Case 6: Strain-Rate Jump
In this example I loaded to 20% engineering strain at a strain rate of 0.01/s, and then switched to a strain rate of 0.1/s. As before, I first generated the stress-strain data using the master material model. I then perturbed the material parameters and then ran the calibration. The following figure shows the final calibration results. The calibrated material parameters match the stress-strain data, but the calibrated parameters are wrong.
Summary
- The following test plans are NOT sufficient to calibrate a Bergstrom-Boyce (BB) material model: (1) monotonic tension at one strain rate; (2) monotonic tension of a “smarter” specimen; (3) load-unload at one strain rate; and (4) a single strain-rate jump.
- The following tests are sufficient: (1) two monotonic tests at two different strain rates; (2) load-relax-load at one strain rate; and (3) multiple load-unload-relaxation cycles.
