A Fractional-Derivative Interpretation of Viscoelastic Rubbers–Part II: Application to Filled and Vulcanized Rubbers Under Large Compression

This paper explains the stress behavior of lled and vulcanized rubbers subject to large deformations using the fractional derivatives proposed in a paper in this series (Fukunaga et al. 2025, CND Vol. 20, 111009, Paper I). The rubber model consists of two fractional- derivative terms and one elastic term arranged in parallel. The orders of two fractional derivatives are α≃ 0.5 and β < α. Of the two fractional-derivative terms, the contribution from the term of order α (the α term) is small for low strain rates. Aside from the α term, the basic parameters are the order, β , the coefficient of the beta term, and the shear modulus of the elastic term. Because the parameters of fractional derivative are few, a deviation from the response of the fractional derivative can be directly interpreted as the effects of the ingredients or a change in state during a course of deformation. At large deformations, the influences of filler and vulcanization are represented by one parameter called the effective thickness, which is a measure of the effective volume fraction of the matrix. A method decomposing these parameters is presented. The fractional-derivative model also suggests differences in the state of rubbers between the loading and unloading phases. The fractional-derivative term weakens or vanishes in the unloading phase. A model with β < 0.2 is consistent with the stress data considered in this paper both in the loading and unloading phases.