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.

The Payne Effect is the reduction in the storage modulus with strain amplitude.

## 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.

The Bergstrom-Boyce model can predict the Payne Effect!

## 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