ABSTRACT
An important trend in the development of (mechanical) constitutive models in general in the incorporation of micro and meso scale phenomena into macroscale models. When dealing with carbon-black filled elastomers this entails a knowledge of the processes occurring in the various constituents and their mutual interfaces. Examples of such processes are elastomer-filler debonding; elastomer saltation over filler particles; filler aggregate breakdown; and matrix, i.e. elastomer degradation. In order to properly model these various phenomena, the individual mechanisms that give rise to them and their driving forces must be understood. For degradation phenomena that occur in the elastomeric or matrix phase of the material, one must have an accurate measure of the stress and strain fields in the elastomer phase as opposed to the corresponding fields averaged over the entire composite. The stress and strain fields are inhomogeneous in such materials even under nominally homogeneous macroscopic boundary conditions. For degradation processes that occur in the bulk portion of the elastomeric phase, the average stress and average strain fields in the elastomeric phase itself can be considered as driving forces depending on the specific process being studied. To the knowledge of the authors in the carbon-black filled elastomer literature, there are hardly any works where the stress and strain fields within the elastomer phase have been explicitly studied using a full field simulation of the micro-structure. The work of Gusev 2006 should be mentioned in this regard. He used
reinforced with particulates to make them a viable engineering material. The most commonly used reinforcing substances are Carbon Black (CB), silica and clay. The composite is processed into its final form by first thoroughly mixing its constituents (i.e. elastomer and reinforcing particulates) along with some additives like sulphur and antioxidants, shaping the mixture into its final form and then vulcanising the shaped mixture. Simple and complex processing techniques exist which can be found in abundance in literature. In this work we focus on CB filled rubber. In case of CB reinforced elastomers, the behaviour of the two independent micro-constituents is well defined in literature. CB is a form of polycrystalline carbon that has a high surface-area-to-volume ratio. It is very stiff and exhibits linear elastic material behaviour. Conventional rubbers are cross-linked amorphous polymers well above their glass transition temperature. They are very soft and can be stretched to several times their original length without breaking. Upon release of the stress, they immediately return to their original length. Their deformation is instantaneous and show almost no creep. The non-linear elasticity (hyperelastic behaviour) of rubbers is predominantly entropy-driven. A more detailed analysis shows that the elastic force originates both from changes in conformational entropy and changes in the internal energy. The latter are normally small and at constant volume relate to changes in conformational energy. To capture the non-linear elastic and nearly incompressible mechanical behaviour of rubber, numerous phenomenological and micro-mechanically motivated models have been proposed in the literature. Steinmann et al. 2012 review the well known representatives of phenomenological and micromechanically motivated models. From the review, it can be clearly noted that the phenomenological models depend on a constant engineering parameter shear modulus (μ ) along with one or more phenomenological parameters that are extracted by fitting these models to experimental results. On the other hand most micro-mechanically motivated models which are based on the concept of freely-jointed chains, depend on the chain density (nc) and the number of chain segments (Nk; often refered to as Kuhn segments). The strain energy density function of such material models for a bundle of chains is as mentioned below:
ψ λ β β βchains
chain = +
⎛ ⎝⎜
⎞ ⎠⎟n N k N ln sinhc k b kΘ (1)
where β λchain k 1 ]N/ and L−1 represent the
inverse Langevin function. The chain density along with the absolute temperature (Θ) and Boltzmann’s
constant (kb) contributes to the definition of the shear modulus (μ = n kc bΘ), which is constant under the assumption of an isothermal process. Therefore, from the available hyperelastic material models for rubber elasticity, it is apparent that the overall non-linear behavior of rubber and rubber like materials depend only on stretches. A typical stress-strain curve of rubber elasticity is shown in Figure 1. It is common knowledge that any reasonable non-linearly elastic model can be represented by a linear-elastic model if only small strains are considered. Using this generalization, we restrict ourselves to small strains in order to attain a Representative Volume Element (RVE) size which is representative of the entire composite (converged RVE) and its homogenized material parameters in a computationally efficient manner. In the next section, the microstructure under consideration has been discussed.
