## Introduction

Calibrating a simple hyperelastic model is easy, most FE packages come with a built-in tool for performing that calibration. Calibrating a more accurate viscoelastic (or viscoplastic) material model is more difficult for two reasons:

- Non-linear viscoplastic material models are often highly non-linear, and optimization algorithms often get stuck in the wrong region of the parameter space.
- Most FE packages do not even come with an easy to use tool for performing the parameter calibration.

The first goal of this article is to show an example of how to quickly calibrate a much more accurate material model than a simple hyperelastic model to real experimental data for an ABR (acrylonitrile butadiene rubber, also called Nitrile rubber, or NBR, or Buna-N).

The second goal is to compare the accuracy of a few different candidate material models for this rubber, and to determine which model is the most accurate.

Here is a simple 3-step MCalibration demo: (1) drag-and-drop the experimental data; (2) Select the type of material model; (3) Select the optimization method. MCalibration will then automatically calibrate a recommended material model for the specified material type.

## Experimental Data Setup

For this study I ran 3 experimental compression tests on the ABR rubber. Two of the tests were constant strain rate compression followed by unloading at true strain rates of -0.1/s and -0.01/s, and one test consisted of compression to a true strain of -0.3 followed by a sinusoidal strain (with an amplitude of 0.02 and frequency of 1 Hz) for 10 minutes, as shown in the figure below.

The three experimental data files were loaded into MCalibration by dragging-and-dropping them. This automatically creates one load case for each experimental test. Note, however, that many times it is necessary to clean up experimental data once it has been read in. Fortunately, MCalibration contains very powerful functions for cleaning up experimental data. To calibrate any material model we also need to specify the bulk response, that is, the bulk modulus or the compressibility (which is typically defined by 2 divided by the bulk modulus). In many experiments we use Digital Image Correlation (DIC) to measure the applied strain. From these experiments we can also extract the transverse strain, which then can be used to calculate the Poisson’s ratio as a function of the applied strain. In this example, the transverse strain was not measured and we therefore assign a target Poisson’s ration of 0.495 using a “Poisson’s Ratio” load case, see figure below.

## Material Model Setup

The next step after importing the experimental data is to select a material model to calibrate. MCalibration support a large number of material models from many different FE solvers. In this article I will study the following material models:

- ANSYS Yeoh hyperelastic
- Abaqus hyperelastic with linear viscoelasticity
- ANSYS Bergstrom-Boyce (BB) model
- PolyUMod Bergstrom-Boyce (BB) model
- PolyUMod BB model with Mullins damage
- PolyUMod Three Network Viscoplastic (TNV) model

## Results: ANSYS Yeoh Hyperelasticity

The Yeoh hyperelastic model is one of my favorite hyperelastic models. It is based on a third order polynomial of the strain energy density in terms of the invariant I_{1}. This hyperelastic model is easy to use and is typically stable. One additional advantage of the Yeoh hyperelastic model is that it can be calibrated to uniaxial data only, since the energy function only depends on the first invariant. The best fit of the Yeoh hyperelasticit model to the ABR is shown in the following figure. The average error (NMAD) between the experimental data and the model predictions is 15.3%. The predictions are obviously really bad since the model cannot predict any of the energy dissipation.

## Results: Abaqus Hyperelasticity with Linear Viscoelasticity

The results in the previous section clearly show that the ABR material tested in this project dissipates significant energy during deformation. This energy dissipation causes the material to behave in a viscoelastic way, and is the main reason the material response cannot be accurately modeled using a basic hyperelastic material model. In an attempt to overcome this issue we will here try Linear Viscoelasticity (LVE), which is a natural extension of hyperelasticity. In the figure below I calibrated a Yeoh hyperelastic model combined with a 5-term Prony series linear viscoelastic model. The best fit of the LVE model is shown in the following figure. The average error between the experimental data and the model predictions is 9.46%. The predicted error in this case is better than for the Yeoh hyperelastic model, but the predicted strain-rate and energy dissipation behavior is not accurately representing the experimental observations. This is a clear sign that the material response is **non-linear** viscoelastic, and therefore one cannot expect a linear viscoelastic model to will work well here.

## Results: ANSYS Bergstrom-Boyce (BB) Model

After showing that a simple hyperelastic or linear viscoelastic material model does not work well, we will try the ANSYS implementation of the Bergstrom-Boyce (BB) model. This is the model that I developed as part of my Ph.D. work at MIT. You can read my whole thesis here. The BB-model is a non-linear viscoelastic material model that is suitable for elastomer-like materials. We can quickly and easily calibrate this model by using the following 3 steps in MCalibration: (1) load in the experimental data; (2) select the ANSYS BB model; and (3) click `Run Calibration`

. The results from the calibration are shown below. The average error of the model predictions is about 7.67%, which is better than for linear viscoelasticity.

## Results: PolyUMod Version of the Bergstrom-Boyce (BB) Model

Many FE codes (e.g. Abaqus, ANSYS, LS-DYNA, and ADINA) have a built-in version of the BB-model. My version of the BB-model is available in the PolyUMod library. The different versions of the BB-model are mostly similar, but there are some small differences between them. Two advantages of the PolyUMod version are that it is available for all FE solvers that PolyUMod supports, and that you can switch between different FE solvers using exactly the same material model. Here, the calibrated results from the PolyUMod BB-model are shown below. The average error is 6.35%.

## Results: PolyUMod Bergstrom-Boyce-Mullins Model

One of the reasons the BB-model is not more accurate than it is for this experimental data set is that the experimental data contains Mullins damage. Typically the Mullins damage is quantified by performing cyclic tests with either constant strain amplitude or with growing strain amplitude. The results from those tests can directly be used to quantify the Mullins damage. Another way to see if a rubber material exhibits Mullins damage is to compare the initial Young’s modulus at small strains to the tangent Young’s modulus right after unloading. In this case, for the ABR material studied here, we see that the unloading slope is steeper than the initial small strain slope of the stress-strain curve. This indicates that the material has undergone Mullins damage. To better capture the experimental data it is therefore useful to use a material model that can capture Mullins damage. In this case I examined the PolyUMod Bergstrom-Boyce-Mullins (BBM) model. The results from calibrating this model are shown in the figure below. The error is about 2.81%, which is **significantly** better than any of the other models what we examined earlier.

## Results: PolyUMod TNV Model

My current favorite material model for rubber materials is the PolyUMod Three Network Viscoplasticity (TNV) model. The TNV model is a multi-network model that supports up to 3 parallel viscoplastic networks. Each network contains an isotropic or anisotropic hyperelastic component, with an optional isotropic or anisotropic flow element. The TNV model also supports multiple advanced failure models that can handle damage, anisotropic response, strain-rate, and triaxiality-dependent failure behaviors. In this case the material response is an isotropic rubber, so the TNV model can be as simple as an two parallel networks, the first can be a Yeoh hyperelastic element with Mullins damage capturing the large strain response of the material, and the second network can be a Yeoh hyperelastic network in series with a power-law flow element. This material model can predict the experimental data with an average error of about 2.51%, which obviously is excellent!

## Summary and Conclusions

The accuracy of the different material models are compared in the figure below. Here are the key points:

- No hyperelastic model can capture the experimental response of the ABR rubber studied in this article. It is a common mistake to think that rubbers and elastomers can be accurately modeling using hyperelasticity. The main problem with hyperelasticity is that it does not predict any viscoelastic effects, but virtually all rubbers and elastomers are viscoelastic!
- It is also interesting to note that linear viscoelasticity (LVE) does not work well here either. The theory of linear viscoelasticity is based on linear viscosity, which is not true for this (and many other) rubbers.
- It is no surprise that the BB-model is much better. I developed the BB-model specifically for predicting the large-strain response of elastomer-like materials.
- It is possible to very accurately model the response of any rubber or elastomer by selecting a suitable material model. Here the correct choice is a BB-model with Mullins damage, or even better, the PolyUMod TNV model which can also capture other types of damage and failure.

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