Editorial Type:
Article Category: Research Article
 | 
Online Publication Date: 01 Mar 2019

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

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Page Range: 2 – 30
DOI: 10.2346/tire.18.470104
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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.

Keywords: creep law; PRF; viscoelasticity; Abaqus; master surface; model calibration
FIG. 1
FIG. 1

Master surface: storage modulus E′ (left) and loss modulus E″ (right) in a wide frequency (freq.) and strain amplitude (range).

FIG. 2
FIG. 2

The original (top) and simplified (bottom) version of the one-dimensional rheological model in parallel rheological framework.

FIG. 3
FIG. 3

Kinematic assumption.

FIG. 4
FIG. 4

Linear (left) vs nonlinear (right) creep law.

FIG. 5
FIG. 5

The first derivative of the linear (LV) and nonlinear creep laws (NLV1 and NLV2) with respect to viscoelastic stress.

FIG. 6
FIG. 6

Different ways to evaluate the complex modulus.

FIG. 7
FIG. 7

Numerical approach to evaluate the complex modulus.

FIG. 8
FIG. 8

Numerical solutaion based on FFT of the stress/strain signals at 0.1% strain amplitude vs direct solution. Stress-strain curves are presented by elliptic hysteresis.

FIG. 9
FIG. 9

(a) Storage modulus E′ in a wide frequency and amplitude range; (b) direct solution approach vs (c) FFT solution approach.

FIG. 10
FIG. 10

(a) Loss modulus E″ in a wide frequency and amplitude range; (b) direct solution approach vs (c) FFT solution approach.

FIG. 11
FIG. 11

Complex modulus of the generalized Maxwell element with two creep process (two Maxwell elements): storage modulus (left) and loss modulus (right).

FIG. 12
FIG. 12

The generalized Maxwell element with two creep process (two Maxwell elements).

FIG. 13
FIG. 13

Measured (meas.) and simulated (sim.) complex modulus of a silica-filled rubber compound in a wide range of frequency and strain amplitude; storage modulus (left), loss modulus (middle), and loss factor (right).

FIG. 14
FIG. 14

Measured (meas.) and simulated (sim.) complex modulus in wide frequency (freq.) range, but at a fixed strain amplitude of 2% at room temperature.

FIG. 15
FIG. 15

Measured (meas.) and simulated (sim.) complex modulus in a wide strain amplitude (amp.) range at three different temperatures, but at a fixed excitation frequency at 10 Hz.

FIG. 16
FIG. 16

Force-displacement response of the hyperbolic-sine creep law and the Prony series model: (left) at low frequency and small strain amplitude, (right) at high frequency and large strain amplitude.

FIG. 17
FIG. 17

Measured (gray dots) and simulated (colored) complex modulus for another compound in a wide range of frequency and strain amplitude: storage modulus (left), loss modulus (middle), and loss factor (right). Vertical black line shows the difference between experiment and simulation.

FIG. 18
FIG. 18

Measured and simulated complex modulus of another filled rubber compound in wide frequency range, but at a fixed strain amplitude of 2% at room temperature.

Contributor Notes

1 Corresponding author. Virtual Development and Computational Mechanics, Continental AG, Jaedekamp 30, 30419 Hannover, Germany. Email: gautam.sagar@conti.de
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