## Constitutive modeling of rubber behavior in a broad frequency domain

Bearbeitung: | Prof. Dr.-Ing. Udo Nackenhorst, M.Sc. Anuwat Suwannachit |

Förderung durch: | This work is supported by German Ministry for Economics within the “Leiser Straßenverkehr 2” program |

Modern passenger car tires are assembled of different rubber blends in order to optimize the overall performance during rolling. Mechanical response of these technical rubber components is usually characterized by their large deformations, hysteretic behavior and damage (Mullins effect) when subjected to quasi-static cyclic loadings. Under harmonic excitations rubber shows its frequency-dependent behavior presented in terms of complex modulus. Despite the fact that a variety of constitutive theories for rubber have been suggested in the past, so far they are only restricted to either quasi-static loadings or high-frequency dynamic analysis. A goal of this project is to introduce a constitutive modeling approach that consistently connects inelastic effects established at low frequencies, e.g. large viscoelastic deformations and damage, with damping behavior in high frequency domain.

The classical Simo’s viscoelastic constitutive model, phenomenologically characterized by a generalized Maxwell solid, is used for describing the viscous dissipation and rate-dependent behavior. These inelastic effects are presented by a set of stress-like internal variables, which are governed by a linear evolution law. The Mullins effect is incorporated by replacing the deviatoric elastic potential in the original viscoelastic model with a pseudo-elastic function proposed by R.W. Ogden and D.G. Roxburgh. The detailed derivations of constitutive equations are presented in the recent publication [Suwannachit&Nackenhorst, 2010]. In figure 1, a good agreement between experimental data of carbon black-filled styrene-butadiene rubber (60 phr) under cyclic uniaxial tension test and computational results can be observed. The extended tube model is used for the deviatoric part of elastic free energy (related to time-infinity response).

Figure 1: Quasi-static uniaxial tension test (carbon black-filled rubber)

For the numerical solution of complex modulus under an arbitrary static pre-deformation, the multiplicative decomposition of material motion into two parts is suggested. One stands for large inelastic deformations and the other one is associated with small vibrations about a certain prescribed configuration. Based on this kinematic assumption a staggered computational strategy is suggested. In the first step a nonlinear computation considering inelastic effects is performed with Newton-Raphson method. The solutions from this step, such as location of particles and inelastic variables, are used as input for the subsequent dynamic analysis, for which the linearized version of the proposed constitutive model is required. The modeling approach is validated by a dynamic test, in which rubber specimen is first driven by 10% strain and subsequently excited by harmonic displacements with 0.02% strain amplitude. A good agreement between experiment and simulation is shown in figure 2.

Figure 2: Experimental and computational complex moduli of tire tread rubber