## Introduction

I often use DMA temperature sweep data to determine how the Young’s modulus depend on temperature. This is of course useful, but also a bit wasteful since only the storage modulus is used and the loss modulus is neglected. In this article I will show how to use both the storage and loss modulus from a DMA temperature sweep test to determine a viscoelastic material model.

## Experimental Data

Temperature sweep data for Hytrel HTR8724 is freely available online. This is the same material that I studied in a recent article about monotonic tension. The goal of this article is to demonstrate how to use this type of experimental data in a material model calibration. As I will show, it is possible to calibrate viscoelastic temperature-dependent material model from the available DMA data.

*Fig 1. Experimental DMA Data.*

## Material Models

In this study I examined a linear elastic material model, and three different linear viscoelastic models. All of these models can capture the experimentally determined storage modulus as a function of temperature data, but they have very different predictions of the loss modulus response.

### Linear Elastic

It is easy to calibrate a temperature-dependent linear elastic material model using MCalibration. Just read in the experimental temperature sweep DMA data, select the material model, and the run the calibration. This model, of course, predicts that the loss modulus is always zero (0). Which is not right.

### Linear Viscoelastic with 1 Prony Term and WLF

Another way to capture the temperature dependence of the storage modulus is to use a linear viscoelastic material model with 1 Prony term and WLF temperature dependence. Figure 3 shows that this model predicted loss modulus is 10X larger than it should be. That is pretty bad!

### Linear Viscoelastic with Prony Spectrum and WLF

Using a linear viscoelastic material model with a full Prony series **spectrum** and WLF temperature dependence significantly improves the ability to match the experimental DMA frequency data. Figure 4 shows that the predictions look really good, except that the predicted storage modulus is not very accurate at large temperatures.

### Temperature Dependent Linear Viscoelastic with Prony Spectrum and WLF

For this Hytrel material, in order to match all the experimental data it is necessary to both make the instantaneous linear elastic response temperature dependent, and to use a Prony series spectrum with WLF temperature dependence. Fig. 5 shows that adding simple temperature dependence to the linear elastic response is very beneficial.

## Predicted Frequency Response

The 4 material models examined above were all able to accurately predict the storage modulus change with temperature, and two of the models were also able to simultaneously predict the loss modulus change with temperature. All of these models, however, have very different predictions of the frequency response at different temperatures.

Fig. 6 show that just having one Prony term gives a very sharp storage and loss modulus transition with frequency. This is why this model over-predicts the peak of the Loss modulus for both frequency- and temperature-sweeps. This model is clearly not accurate for this material.

*Fig 6. Frequency sweep predictions from a 1-term Prony series with WLF material model.*

Fig. 7 shows that a Prony spectrum model can have a very wide transition of the storage and loss modulus with frequency. In fact, the spectrum is wider than expected, and it is unlikely that the real material would have the same max storage and loss modulus for all temperatures. For these reasons, this model is likely not that accurate either.

*Fig 7. Frequency sweep predictions from a Prony spectrum model with WLF.*

Fig. 8 shows the predicted storage and loss modulus as a function of frequency for the viscoelastic model with a temperature-dependent modulus and a Prony spectrum with WLF. In this case the predictions look qualitatively as expected. Note that it would have been useful to have actual DMA frequency data for the material model calibration and/or validation.

*Fig 8. Frequency sweep predictions from a temperature-dependent elastic model with Prony spectrum and WLF.*

## Predict Monotonic Tension Response

Fig. 9 shows the predicted uniaxial tension response at room temperature at 6 different strain rates. Clearly the predictions from the linear viscoelastic models look more realistic than the basic temperature-dependent elastic model.

*Fig. 9. Predicted monotonic uniaxial tension response.*

## Summary

- It is easy to calibrate a temperature-dependent elastic material model to DMA temperature sweep data. This type of model, however, always predicts that the loss modulus is zero (which is not true).
- It is also possible to add linear viscoelasticity with WLF to the temperature-dependent elastic model in order to get a more accurate model.
- To get the most accurate results I recommend also perform some temperature sweep DMA tests.