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

Abstract

Abstract. This paper explains the stress behavior of filled and vulcanized rubbers subject to large deformations by using the fractional derivatives proposed in a paper in this series (Fukunaga et al., 2025, “A Fractional Derivative Interpretation of Viscoelastic Rubbers. I. Thermodynamically Consistent Fractional-Derivative Models for Finite Strain,” ASME J. Comput. Nonlinear Dyn., 20(11), p. 111009, Paper I). The proposed 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 β term, and the shear modulus of the elastic term. Because the number parameters of fractional-derivative model is small, 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 the 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 for 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.