## Introduction - Best Material Models

I performed a careful study to determine the best material model for a soft copolyester. Recall that thermoplastic copolyester elastomers (TPC) is a class of thermoplastic elastomers (TPEs) that combine many of the properties of both thermoplastics and rubbers. Examples of copolyesters include:

- Tritan from Eastman (stiff, strong thermoplastic)
- Arnitel from DSM is a thermoplastic elastomer (TPE) with high temperature resistance, good processability and excellent elastic behavior. Arnitel belongs to the TPE subcategory of thermoplastic copolyesters (TPC).

The study is based on the same data as in my article on stress relaxation vs cyclic loading. All material model calibrations were performed using MCalibration.

Goal: Find a material model that works for all experimental data for the TPE

## Results

### Linear Elastic

Note: linear elasticity does not create a linear engineering stress-strain curve at finite strains. Max error in the predicted stress: 3.18 MPa.

### Yeoh Hyperelastic

Hyperelasticity should not be used beyond yield. The model predictions look odd (in order to best match the experimental data).

### Yeoh Hyperelastic with Linear Viscoelasticity

Linear viscoelasticity should not be used beyond yield!

### Johnson-Cook

The Johnson-Cook plasticity model is a commonly used plasticity model. It contains both an equation-based hardening rule and strain-rate dependence. The average error in the model predictions is 7.13%, the max error in the stress predictions is 2.16 MPa.

### Abaqus Elastic-Plastic-Combined Hardening

This is a 5 back-stress network kinematic hardening plasticity model with the following Abaqus commands:

`*Elastic`

`*Plastic, hardening=combined, data type=parameters`

This model is also sometimes called the Chaboche model, or the Frederick-Armstrong model. The best fit of this model has an average error of 6.54% and a max error in the predicted stress of 1.43 MPa. These values are surprisingly good.

### Ansys MISO Plasticity with Creep

The multilinear isotropic hardening plasticity model combined with creep is based on the the following Ansys commands:

`MP, EX, matid, 44.8427650012,`

`MP, NUXY, matid, 0.4,`

`TB, PLASTIC, matid, 1, 10, MISO`

`...`

`TB, CREEP, matid,1,, 1`

`..`

The best fit of this model has an average error of 4.41% and a max error of the predicted stress of 4.10 MPa.

The plasticity models are not terrible, but we can get significantly more accurate predictions by switching to a multi-network viscoplastic material model.

### PolyUMod Bergström-Boyce (BB) Model

The Bergström-Boyce (BB) model is described in detail in this post. The best fit of this model has an average error of 2.18%, which is fantastic. The max error of the predicted stress is 5.08 MPa, which occurs during unloading. It is well known that 2-network viscoplastic models tend to overpredict the residual strain of thermoplastic elastomers (TPEs) and thermoplastics. This is not surprising since I developed the BB-model for elastomers.

### PolyUMod Bergström-Boyce (BB) Model with Mullins Damage

The best fit of this model has an average error of 2.33%, and a max error of the predicted stress of 4.94 MPa. In this case, activating Mullins damage does not significantly improve the accuracy of the model predictions.

### Abaqus PRF (“PRF2YP”)

The Abaqus Parallel Rheological Framework (PRF) model with 2 Yeoh hyperelastic networks and power-law flow is a material model with a structure that is similar to the BB-model. The best fit of this model has an average error of 3.75%, and a max error of the predicted stress of 4.17 MPa. These values are similar to the PolyUMod BB-model.

### Ansys Bergstrom-Boyce (BB)

The Ansys implementation of the BB-model is very similar to the PolyUMod version. The best fit of the Ansys BB-model has an average error of 2.85%, and a max predicted stress of 4.98 MPa.

### Ansys Bergstrom-Boyce (BB) Model with Mullins Damage

Activating Mullins damage with the Ansys BB-model does not significantly improve the model predictions. The best fit of the model has an average error of 2.43%, and a max predicted stress of 4.98 MPa.

### PolyUMod Three Network (TN) Model

The PolyUMod Three Network (TN) model is a versatile and accurate material model for most polymers, including thermoplastic elastomers. The best fit of this model has an average error of 2.12%, and a max error of the predicted stress of 0.86 MPa. **These results are quite good, and certainly better than any other model that we been discussed so far.**

### PolyUMod Three Network Viscoplasticity (TNV) Model without Mullins Damage

The PolyUMod TNV model is currently my favorite model. It is even more accurate than the TN model, and has additional features (e.g. failure, anisotropy). The best fit of this model has an average error of 1.57%, and a max error of the predicted stress of 1.22 MPa. Both of these values are very good.

### PolyUMod Three Network Viscoplasticity (TNV) Model with Mullins Damage

The best fit of the TNV model with Mullins damage has an average error of 1.45%, and a max error of the predicted stress of 0.87 MPa. **This accuracy is outstanding, the best of any model that is investigated in the study! This model is the winner.**

### Abaqus PRF (“PRF3YPM”)

The Abaqus Parallel Rheological Framework (PRF) model with 3 Yeoh hyperelastic networks, power law flow elements, and Mullins damage is reasonably good, but not as accurate as the PolyUMod TN or TNV models. This may be surprising since they are all 3 network viscoplastic models with lots of similarities. The reason for the difference is that the TN and TNV models have slightly different governing equations for the flow elements. The best fit of the PRF model has an average error of 2.2%, and a max error of the predicted stress of 3.19 MPa. The max error of the predicted stress occurs during unloading, and is concerning.

The Abaqus Parallel Rheological Framework (PRF) model with 4 hyperelastic networks with power law flow elements provide predictions that are just slightly better than the corresponding 3 network PRF model. The best fit of the PRF model has an average error of 1.67%, and a max error of the predicted stress of 3.11 MPa. This max error in the stress is also concerning in this case!

## Summary and Conclusions

The following figure compares the average error of all material models that were were part of this study. Clearly the linear elastic (LE), Yeoh hyperelastic, and the linear viscoelastic models are much less accurate than the other models, and should not be used for this material.

If I remove those models then the average errors of the remaining models look as follows (red=plasticity, green=2 network viscoplastic, blue=3 network viscoplastic, pink=4 network viscoplastic). We see that all viscoplastic models outperform the plasticity models. The average errors of all viscoplastic models are very low. The model with the best (lowest) average error is the PolyUMod TNV model.

Another and even better way of plotting the data is shown below. In this figure the error bars include both the average error and the max error of the model predictions. It is clear from this figure that many of the viscoplastic models are able to accurately predict the average response of the material, but only the PolyUMod TN and TNV model also have a low max error. The max error in this case typically occurred after unloading and is an indication of a poor prediction of the residual strain. It is particularly interesting to see that the max error of the BB-type models and the PRF models is about 40%. Clearly not good predictions.