Thermal Fluctuations and Rubber Elasticity

The effects of thermal elastic fluctuations in rubbery materials are examined. It is shown that, due to their interplay with the incompressibility constraint, these fluctuations qualitatively modify the large-deformation stress-strain relation, compared to that of classical rubber elasticity. To leading order, this mechanism provides a simple and generic explanation for the peak structure of Mooney-Rivlin stress-strain relation and shows good agreement with experiments. It also leads to the prediction of a phonon correlation function that depends on the external deformation.

Figure 1

The Mooney-Rivlin plot of the Mooney stress $(df/d\lambda )/(\lambda -{\lambda }^{-2})$ for a uniaxial shear deformation, Eq. (2), versus $1/\lambda$. While the classical theory predicts a horizontal line, real rubbery materials universally exhibit a pronounced peak feature. Squares: data from Xu and Mark [13] (unit: ${10}^5\mathrm{Pa}$). Top curve: our theory fit that incorporates lowest-order phonon fluctuations on scales beyond the crosslink spacing, using Eq. (17). Fitting parameters: $B=\infty$, ${\mu }_0=8.92$, ${\mu }_1=7.10$. Also plotted are the curves for finite bulk moduli $B$, which are calculated using Eq. (18) and the same fitting parameters ${\mu }_0$ and ${\mu }_1$.