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# How to Model the Gough-Joule Effect in Rubber Seals

## Introduction

The Gough-Joule Effect is a phenomenon that makes some rubber products shrink if they are under load and the temperature is increasing.  This effect can be important in various applications, for example in rubber seals that are pre-loaded at room temperature and then exposed to elevated temperatures during normal use. In this article I will discuss what causes this effect and how you can model the Gough-Joule effect in a FE simulation. Figure 1. James Joule (1818 – 1889) was an influential physicist that helped discover the first law of thermodynamics.
John Gough (1757 – 1825) lived about 100 years before Joule. John Gough was blind.

## Experimental Observation #1  Just a spring. No force. Not much is happening…

Just a spring. No force. Heat it up. Small thermal expansion.

## Experimental Observation #2   Just a spring. A force is applied. The spring elongates.

Just a spring. A force is applied. Heat it up. Most spring materials elongate!

Just a rubber spring. A force is applied. Heat it up. Some rubber springs shrinks! This is the Gough-Joule Effect!

## Why Care About the Gough-Joule Effect? If a rubber seal is mounted in a deformed state then the rubber shrinkage at high temperatures can have serious consequences for the seal performance. For example, the Gough-Joule effect can cause a rotary seal to bind if the seal shrinks due to overheating.

Note: The Gough-Joule effect is caused by the odd temperature dependence that occurs in some rubbers. Specifically, for some rubbers the deformation resistance is mainly caused by entropic effects, which causes the Young’s modulus to increase with the absolute temperature. That is something that James Joule did not know!

## How to Model the Gough-Joule Effect

To demonstrate how to model the Gough-Joule effect I will start with some experimental data for an unfilled silicone rubber (see Figure 2). The data consists of a single uniaxial tension load cycle at 4 different temperatures. As is shown here, unfilled silicone typically exhibits only a small amount of hysteresis. What is more interesting is that the stiffness of the material increases with temperature. Figure 2. Cyclic tension data for an unfilled silicone rubber [Polymer Testing, 2013, 32 (2), 492-501].

To model the response of the silicone rubber I used the PolyUMod TNV model. This material model is an advanced viscoplastic model that can accurately predict the response of most polymers. Here I selected 2 networks to represent the material behavior: (1) A temperature-dependent Yeoh hyperelastic network; and (2) a Yeoh hyperelastic network with flow cessation and temperature dependence. I typically recommend using experimental data with stress relaxation or multiple strain rates in order to use a viscoplastic material model. But in this case I think this model is good enough for demonstration purposes. Figure 3 shows the material model selection in MCalibration. Figure 3. TNV material model structure.

Figure 4 compares the experimental data to the predictions from the calibrated temperature-dependent material model. The error in the predicted stress-strain curves is 2.2%. Not too bad. Figure 4. Predictions from the final TNV material model.

It is easy to use MCalibration to check if this material model predicts the Gough-Joule effect. Just create the virtual load case shown in Figure 5. In this load case the sample is first uniaxially loaded until the engineering stress reaches a value of 0.5 MPa. The stress is then held constant for 60 seconds while the temperature is increased from 20°C to 120°C. Figure 5. Virtual load case in MCalibration to testing the Gough-Joule effect.

The resulting stress-strain prediction is shown in Figure 6. The figure shows that the strain decreases during segment 2 in which the temperature is increased. In other words, the sample shrinks at elevated temperatures when under load. This is the Gough-Joule effect! Figure 6. Virtual load case illustrating the Gough-Joule effect.

## Summary of the Gough-Joule Effect

• Only occurs for some elastomers
• The reason for the Gough-Joule effect is that the modulus for some elastomers increases with temperature due to entropic effects
• Is easy to model in a FE simulation if you use an accurate temperature-dependent material model (like the PolyUMod TNV model)