## Introduction

There are 2 ways to create a temperature-dependent material model:

## Approach 1: Interpolate Material Parameters

For simple material models it is possible to simply linearly interpolate each material parameter based on the temperature. One can use as many temperatures as needed in order to capture the experimentally observed behavior.

It is possible to use this interpolate-each-parameter approach also for more non-linear material behaviors. The figures below show calibrations of the TNV model to experimental data for PETG. And yes, this material can indeed behave in this “odd” way in compression. All I did here was to use MCalibration to calibrate a three-network TNV material model to the data for each temperature. If you look at the parameters carefully, you will see that even though the response at 60 deg C is softer than at 25 deg C, some of the parameters are unexpected. For example, the Yeoh C10 parameter for the second network is higher at 60 deg C than at 25 deg C.

In theory it is possible to get odd/bad predictions when each parameter is simply interpolated based on the temperature. For this reason, I recommend calibrating the lowest temperature model first, and then use the results from that calibration when calibrating a model for the next temperature. This usually makes each parameter set somewhat similar (which is useful when each parameter is interpolated).

Most FE solvers support material parameter interpolation of most material models, but there are exceptions to this. For example. Abaqus does not allow for parameter interpolation of user-material models. Due to this limitation in Abaqus, the PolyUMod library has a Multi-Temperature model framework that handles the parameter interpolation.

### Example: Ansys-Style Linear Parameter Interpolation

Ansys has a brilliant way to handle material parameter interpolation for the PolyUMod library. All that is needed is to specify the `TBTEMP`

command before providing the parameters for that temperature. Here’s an example of the APDL syntax:

` ````
```TB, USER, matid, 1, 62
TBTEMP, 25
TBDATA, 1, 29
... all parameters for the first temperature here
TBTEMP, 60
TBDATA, 1, 29
... all parameters for the second temperature here

The following figures show the Ansys predictions of the Mises stress using this linear interpolation approach of the calibrated TNV model. The FE model that I used here is a simple dogbone-shaped test case. As expected, the predictions are temperature dependent.

### Example: PolyUMod Multi-Temperature Model

The PolyUMod Multi-Temperature model framework was developed in order to enable parameter interpolation when using other FE solvers than Ansys. The theory of this model is available in Chapter 27 of the PolyUMod User’s Manual. It is also possible to calibrate a Multi-Temperature model using a PolyUMod template in MCalibration. The calibration results for the PETG example from above are shown in the following figure.

The following figures show the Ansys predictions of the Mises stress using this linear interpolation approach of the calibrated TNV model. The predictions are similar to the Ansys linear interpolation model.

## Approach 2: Equation-Based Temperature Models

Most FE solvers do not have material models with equation-based temperature dependence. One exception to this is the WLF and Arrhenius models for viscoelasticity. The PolyUMod library, however, has multiple models with equation-based temperature models. The TNV model, for example, supports both linear- and tanh-based temperature models. In this example I calibrated a three network TNV model with linear temperature dependence to the experimental data for the PETG. The calibration results are shown in the following figure.

The following figures show the Ansys predictions of the Mises stress using this TNV model with equation-based temperature dependence. The predictions are similar to the Ansys linear interpolation model.

## Summary

There are 2 ways to make a material model temperature dependent:

**Linear interpolation of parameters:**- Easy to use (when available).
- Can give odd/bad model predictions if the material model parameters change in unexpected ways between temperatures.

**Equation-based temperature-dependent material models:**- Not many available natively.

Also note that the PolyUMod library supports both interpolation and equation-based temperature-dependent models.