On the Development of Creep Laws for Rubber in the Parallel Rheological Framework

It is widely known that filler-reinforced rubber material in tires shows a very complicated material behavior when subjected to cyclic loadings. One of the most interesting effects for rolling tires is the nonlinear rate-dependent behavior, which is implicitly linked to the amplitude dependency of dynamic stiffness (Payne effect) at a given frequency and temperature. This effect, however, cannot be described by a conventional linear viscoelastic constitutive law, e.g., the Prony series model. Several nonlinear viscoelastic material models have been proposed in the last decades. Among others, Lapczyk et al. (Lapczyk, I., Hurtado, J. A., and Govindarajan, S. M., “A Parallel Rheological Framework for Modeling Elastomers and Polymers,” 182nd Technical Meeting of the Rubber Division of the American Chemical Society, Cincinnati, Ohio, October 2012) recently proposed a quite general framework for the class of nonlinear viscoelasticity, called parallel rheological framework (PRF), which is followed by Abaqus. The model has an open option for different types of viscoelastic creep laws. In spite of the very attractive nonlinear rate-dependency, the identification of material parameters becomes a very challenging task, especially when a wide frequency and amplitude range is of interest. This contribution points out that the creep law is numerically sound if it can be degenerated to the linear viscoelastic model at a very small strain amplitude, which also significantly simplifies model calibration. More precisely, the ratio between viscoelastic stress and strain rate has to converge to a certain value, i.e., the viscosity in a linear viscoelastic case. The creep laws implemented in Abaqus are discussed in detail here, with a focus on their fitting capability. The conclusion of the investigation consequently gives us a guideline to develop a new creep law in PRF. Here, one creep law from Abaqus that meets the requirements of our guideline has been selected. A fairly good fit of the model is shown by the comparison of the simulated complex modulus in a wide frequency and amplitude range with experimental results.