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Non linear rheology of model filled elastomers

Aurélie Papon, Hélène Montes, François Lequeux and Laurent Guy

Submitted to large sinusoidal strains, filled elastomers show not only a decrease in their storage modulus - the Payne effect, but also a non linear behavior - their response is not sinusoidal anymore and involves strain-stiffening. We show in this study that the two effects can be separated thanks to Large Amplitude Oscillatory Shear experiments.

The introduction of solid particles such as silica or carbon black in elastomer matrix is a very standard procedure in the tire industry. It greatly improves the mechanical properties of the rubber but also introduces a non linear behavior.
One example of this complex behavior is the well-known Payne effect : whereas unfilled elastomers exhibit a constant elastic modulus for strain amplitudes up to 100% during cyclic strain experiments, filled rubbers show a significant decrease in their elastic modulus for strain amplitudes typically from a few percents. It is often assumed that in this strain domain, filled elastomers exhibit a quasi-harmonic behavior. It means that under sinusoidal solicitation, they have a quasi-sinusoidal response.

We show that for our model filled samples however the non linearity in the response is significant : a large sinusoidal strain gives rise to a non sinusoidal stress. We analyzed the stress signal in two steps : first the decomposition into elastic and viscous parts and then the extraction of a linear and a non-linear contribution in each part. The modulus given by the rheometer corresponds to the sum of the two components. The non linear contribution is not negligible from strain amplitudes of around 10% in our samples.

We show that the strain-stiffening is correlated with the Payne effect but also most probably has its origins in the finite extensibility of the polymer chains confined between solid particles, where the strain is larger. We also point out that the samples behave differently whether the grafting agent used creates a covalent bond between the silica and the polymer or not.

A. Papon, H. Montes, F. Lequeux and L. Guy. Non linear rheology of model filled elastomers, Journal of polymer science Part B : Polymer physics, in press.

Elastic stress as a function of strain
In the linear regime, the elastic stress plotted as a function of the strain inside one cycle of solicitation is a straight line with a slope G’. Here we can see a deviation from the straight line : there is an intra-cycle strain-hardening, similar to a finite extensibility phenomenon. Moreover, the slope at the origin is decreasing with the strain amplitude : this is the Payne effect.
Storage modulus decomposition into a linear and a non linear contribution
Fitting the elastic stress as function of strain with a polynomial function, we can extract a linear and a non-linear contribution in the storage modulus. From strain amplitudes of 10%, the two contributions are superimposed and the rheometer is only giving the sum of the two.