# Modeling PET Through Tg

## Introduction

The mechanical response of thermoplastics changes A LOT with temperature, especially for temperatures close to the glass transition (Tg). For this reason, most engineers try to stay away from performing FE simulations of thermoplastics at these temperatures. In this article I will show 2 ways to create accurate non-linear temperature-dependent material  models also for temperatures close to the glass transition.

## Experimental Data

In this study I will use experimental data from R.B. Dupaix and M.C. Boyce [Polymer 46 (2005) 4827-4838] for PETG. They tested PETG in uniaxial tension at 3 different strain rates and 5 different temperatures. Figure 1 summarizes their data.

Figure 1. Experimental tension data for PETG.

## Method 1: All-At-Once Calibration

One way to simulate the mechanical response of the thermoplastic is to use an advanced temperature-dependent material model that is based on equations that have been designed for this class of materials. One such model is the PolyUMod TNV model. In this application I used the TNV model settings shown in Figure 2. Each of the 2 networks use a hyperelastic model with the following equation for the temperature-dependence (see the PolyUMod User’s manual for more info about the equations):

$$f_{\theta} = \displaystyle\frac{1}{2} (f_g + f_r) – \frac{1}{2} (f_g – f_r) \cdot \tanh\left[ \frac{\theta – \theta_g}{\Delta\theta}\right] + X_g \cdot \frac{\theta – \theta_g}{\Delta\theta}$$

The model also uses  a similar equation to specify how the flow resistance changes with temperature.

Figure 2. TNV model structure (screenshot from MCalibration).

The calibration results from this model are shown in Figure 3. The amount of curves in this figure is a bit overwhelming, but when studied more closely it is clear that the predictions are pretty good. All major trends are captured correctly. The average error in the model predictions is about 15%, which is OK but perhaps not great.

Figure 3. Predictions from a temperature-dependence TNV model.

## Method 2: Calibrate to Each Temperature

Another approach is to calibrate a material model to each temperature individually, and then combine the individual material models to a master model (which is temperature dependent).  In this case I used 2 parallel TNV network (see Figure 4). The results from the individual calibrations are shown in Figure 5.

Figure 4. TNV model structure.

Figure 5. TNV model predictions for each individual temperature.

The individual calibrations can then be combined into a temperature-dependent master model using the PolyUMod Multi-Temperature Model. This model framework takes the parameter sets for each temperature, and the corresponding temperatures, and then interpolates the parameters as needed during a FE simulation. The results from the final master model are shown in Figure 6.

Figure 6. Predictions from the temperature-dependent TNV model.

## Summary

• It is possible to simulate the response of a thermoplastic through the glass transition temperature.
• It is necessary to have good experimental data.
• Need an accurate material model (like the PolyUMod TNV model).

### Linear Viscoelasticity – Part 6 – Rheological Model

The integral equation form of linear viscoelasticity is identical to a rheological model with parallel spring-and-dashpot networks.

### Smart Mechanical Testing of Polymers

Smart mechanical testing is about how to extract as much info as possible using a small number of experiments.