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

Abstract

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.

Constitutive rubber model suitable for rolling resistance simulations of truck tyres

“…The Prony series was chosen due to its simplicity and lack of amplitude dependency, although many more complicated viscoelastic models do exist such as Bergström-Boyce, 26 hyperbolic-sine law, strain hardening power law and power law. 32 The mentioned nonlinear viscoelastic models are modifications of linear viscoelasticity and can be reduced to the Prony series with certain input parameters. 32 The simplicity of the Prony series and strain amplitude independency for the storage and loss modulus are considered to be an advantage since the frequency stiffening is decoupled from the amplitude dependency of the storage and loss modulus.…”

“…The usage of nonlinear viscoelastic models in a PRF model increases the model complexity and the meaning of the parameters become less clear, which makes parameter identification challenging. Sagar et al 32 gave two recommendations on how to design creep laws for easier parametrisation: at small strains the nonlinear creep laws should reduce to linear viscoelasticity and the creep laws second derivative at small strains with respect to viscoelastic stress has to be nonnegative. The hyperbolic-sine law fulfils these guidelines with a special condition n = 1 .…”

“…22,27 Much less attention has been given to modelling the Fletcher-Gent effect in truck tyre rolling resistance simulations. Nonlinear viscoelasticity, 22,25,27 fractional derivatives 28 or various parallel rheological frameworks 22,[30][31][32][33][34] are examples of popular models to simulate the Fletcher-Gent effect.…”

“…The hyperbolic-sine law fulfils these guidelines with a special condition n = 1. 32 Nevertheless, multiple nonlinear models in parallel can create additional bumps and non-smoothness to hysteresis loops for some frequencies and amplitudes. Highly nonlinear PRF models with unclear material constants require stress-strain or storage/loss modulus data as an input for parameter optimisation to generate material parameters for design and sensitivity studies.…”

Creep of particle and short fibre reinforced polyurethane rubber

“…There has long been interest in creep of rubbers (Plazek 1966;Ngo and Valdes 2007;Oman and Nagode 2014;Sagar et al 2019), partly because rubbers or rubberized materials are commonly used in applications such as seals, damping systems etc, in which they are subjected to prolonged loads at (relatively) high temperatures (Derham 1973;Derham and Thomas 1977;Lee and Rivin 1996;Luo 2016). Furthermore, while some studies have focused on pure rubbers, much industrial usage relates to filled rubbers, so there is also interest (Ngo and Valdes 2007;Oman and Nagode 2014;Chen and Cheng 1997;Daver et al 2016) in (creep of) composite materials based on rubber matrices.…”