# Payne Effect: Theory and Modeling

## Introduction to the Payne Effect

The Payne Effect is a phenomenon that occurs in almost all rubbers. It was discovered by Fletcher and Gent [Trans. Inst. Rubber Ind., 29, 266 (1953)] and analyzed in detail by A. R. Payne in his famous paper from 1962 [J. Appl. Polymer Sci, 19, 57-63, 1962]. The figure to the right shows that the total and storage modulus is almost constant for strains below about 10-4, and then drastically drops as the strain amplitude increases. In summary:

• The softening is stronger in highly filled rubbers, and mainly occurs at small deformations.
• The Payne softening is often attributed to filler-filler interactions.
• The Payne effect cannot be modeled using Linear Viscoelasticity.

## Linear Viscoelasticity

A linear viscoelastic material model automatically predicts a storage modulus that is monotonically increasing with frequency, from an initial low value to a final high value. A linear viscoelastic model also predicts a peak in the loss modulus at a frequency that is in the middle of transition of the storage modulus. The 2 figures below show how it looks with the selected Prony series values. Note that a hyperelastic model with a Prony series can be selected to match any experimental data for the storage modulus, or any experimental data for the loss modulus, but there is no guarantee that a linear viscoelastic model can match both the storage and loss modulus data at the same time.

The response of a linear viscoelastic material model when exposed to strain amplitude sweeps is less exciting. As shown in the 2 figures below, linear viscoelasticity predicts that both the storage and loss moduli are constant with strain amplitude. Which clearly is not true for most rubbers, and linear viscoelasticity clearly cannot predict the Payne effect.

## Non-Linear Viscoelasticity (Bergstrom-Boyce Model)

Since linear viscoelasticity cannot predict the Payne Effect, are there any better options? Yes, of course, the answer is to use a non-linear viscoelastic material model, such as the Bergstrom-Boyce model.

The Bergstrom-Boyce model is a two-network model: one network gives the equilibrium hyperelastic response, and one network gives the non-linear viscoelastic response. The model is non-linear viscoelastic since the dashpot response is non-linear:

$$\dot{\gamma}^v = \dot{\gamma}_0 \left( \overline{\lambda}^v -1 + \xi \right)^{C} \left[ \frac{\tau}{\tau_{base}} \right]^m.$$

What is cool about this flow equation is that if C=0 and m=1, then the dashpot becomes linear viscoelastic, and the Bergstrom-Boyce model  becomes a 2-term linear viscoelastic model.

Now let’s look at the response of the Bergstrom-Boyce model when exposed to a frequency sweep. The 2 figures below show that the predicted dynamic response is very similar to the linear viscoelastic model when exposed to a frequency sweep. It looks very good.

When exposing the Bergstrom-Boyce model to a  strain amplitude sweep the response still look very nice! The storage modulus decreases with strain amplitude, and the loss modulus undergoes a peak. Very cool! The reason the Bergstrom-Boyce model is better is due to the non-linear viscoelasticity introduced by the C and m parameters.

## Bergstrom-Boyce Model when Exposed to Frequency Sweeps

The following figures show how the 4 control parameters in the Bergstrom-Boyce model influence the dynamic response of the model under a frequency sweep. In each figure I kept all material parameters the same, except one parameter that is given 3 different values. In MCalibration this is called a Parameter Sweep.

## Bergstrom-Boyce Model when Exposed to Strain Amplitude Sweeps

The following figures show how the 4 control parameters in the Bergstrom-Boyce model influence the dynamic response of the model under strain amplitude sweeps. In each figure I kept all material parameters the same, except one parameter that is given 3 different values. In MCalibration this is called a Parameter Sweep.

## Payne Effect Summary

• Occurs in almost all rubbers, mainly at small strains
• The effect is stronger in highly filled rubbers
• Cannot be predicted using linear viscoelasticity
• Can be predicted using a non-linear viscoelastic model, such as the Bergstrom-Boyce, or other multi-network model