Microstructure-Based Modelling and FE Implementation of Filler-Induced Stress Softening and Hysteresis of Reinforced Rubbers

Reinforcement of rubber by nanoscopic fillers like carbon black introduces strongly nonlinear mechanical effects like stress softening an hysteresis. In order to predict the mechanical behaviour of a rubber component, these effects should be understood on a micromechanical level and integrated into a mathematical model. One advantage of a microstructure-based compared to a phenomenological approach is that material parameters are physical quantities which can be obtained by fitting of "multihysteresis" stress-strain tests.

The present model refers to the mechanical behaviour of filled rubbers loaded quasi-statically up to large strains. It represents an extension of the previously introduced "dynamic flocculation model" describing filler-induced stress softening and hysteresis [1-3]. The first effect is modelled by hydrodynamic reinforcement of rubber elasticity due to strain amplification by stiff filler clusters. Under stress, these clusters can break and become soft, leading to deformation of larger parts of the volume. This causes stress softening by decreasing strain amplification factor (expressed as an integral over the "surviving", hard, section of the cluster size distribution). The second effect is attributed to cyclic breakdown and re-aggregation of softer, already damaged filler clusters. When deformation cycles are not closed (as occurs during an arbitrary deformation history) not all of the soft clusters are broken at the turning points of a cycle. For these "inner cycles" additional elastic stress contributions of clusters are taken into account.

In a microstructure-based model the dependence of the material parameters on time and temperature can be derived from physical considerations. We attribute the temperature-dependent behaviour above the glass transition to an Arrhenius-activated strength of the filler-filler bonds (glassy polymer bridges of some nanometers in thickness) [2]. Considering separate activation energies for virgin and damaged bonds, only two additional parameters are needed to cover the dependence on temperature. Similarly, the time-dependent behaviour of the filled rubber is modelled by considering the visco-elasticity of the filler-filler bonds.

The uniaxial model has been generalized for three-dimensional stress states using the concept of representative directions - the basic idea being that the stress state at a given material point can be approximated by the response of a number of uniaxially loaded "fibres". The resulting 3D-model was implemented into the Finite-Element-Method (FEM) [4], and some examples, including a rolling rubber wheel, are shown.

Fair agreement between measurement and simulation is obtained in the case of filled rubber specimens, loaded along various uniaxial deformation histories. The 3D-generalization of the extended dynamic flocculation model is able to simulate inner cycles in various deformation modes even closer to the experiment than the original model.

[1] Klüppel, M., Adv. Polym. Sci., 164, pp. 1-86 (2003)

[2] Klüppel, M., Meier, J., Dämgen, M., in: Constitutive Models for Rubber IV", Austrell & Kari, Eds., Taylor & Francis, London (2005)

[3] Lorenz, H., Klüppel, M., in: Constitutive Models for Rubber VI, G. Heinrich et al., Eds., Taylor & Francis, London (2010), p. 423 ff

[4] Freund, M., Ihlemann, J., in: Constitutive Models for Rubber VI, G. Heinrich et al., Eds., Taylor & Francis, London (2010), p. 21 ff