Strain-Dependent Dynamic Properties of Filled Rubber: A Non-Linear Viscoelastic Approach Based on Structural Variables

This essay deals with a general framework based on the phenomenological theory of non-linear thermoviscoelasticity to represent the characteristic strain dependence of dynamic moduli of carbon black-filled vulcanizates (Payne effect). By virtue of thermodynamical arguments we develop a one-dimensional model consisting of non-linear springs and damping elements. We introduce viscosity functions depending not only on the temperature but on other variables besides. They can be related to the current state of the material's microstructure. Under dynamic loads and stationary conditions, these variables become approximately constant so that the model can be simplified. One of the main results is that the reduced set of equations becomes comparable to linear viscoelasticity but the structural variables imply a dependence of the viscosities on the deformation amplitude. It follows from our theory that the amplitude-dependent parts of storage and dissipation modulus are not independent of each other, as frequently assumed. Both moduli depend in a characteristic manner on the strain amplitude. If we assume, for example, a decreasing power-law relation between viscosities and structural variables, the theory reduces to the widely-accepted G′(Δε) and G′″(Δε) models proposed by Kraus and substantiated by Huber and Huber et al. If we introduce an additional mechanism of exponential-type we obtain the improved G′″(Δε) model developed by Ulmer. Under transient conditions with varying strain amplitudes, as studied experimentally by Wang et al., the model describes thixotropic recovery effects of the moduli as well as more complicated memory phenomena. Under further assumptions we can apply the frequency/temperature shift principle in combination with a frequency/amplitude shift principle to determine the material parameters. Since the model is based on rheological elements, its compatibility with the second law of thermodynamics is ensured.