A network evolution model for the anisotropic Mullins effect in carbon black filled rubbers

Abstract

a b s t r a c tTo the best of our knowledge, there are no constitutive models that properly describe experimental data on anisotropy of the Mullins effect. In this paper, such a micro-mechanical model is proposed for carbon black filled rubbers. The model describes the deformation induced anisotropy and permanent set as well. Damage of the polymer-filler network is considered as a consequence of chain sliding on or debonding from aggregates. In contrast to previous works on anisotropy of the Mullins effect we do not introduce any phenomenological damage function. Damage in different directions is governed by a network evolution concept which describes the changes in the inter-aggregate distribution of polymer chains. The model includes a few number of physically motivated material constants and demonstrates good agreement with own experimental data on subsequent uniaxial tensions in two orthogonal directions.

Network evolution model: thermodynamics consistency, parameter identification and finite element implementation

“…In the network evolution model there are seven physically motivated material parameters, {κ, R, n max } are directly influenced by the concentration of filler and ν, n c ,Ñ 0 ,Ñ c are independent of the reinforcement [1]. Therefore, we firstly determined all seven material parameters of the model by fitting to the experimental results on uniaxial tension of 60 phr CR in x-direction.…”

“…The PP network is assumed to be responsible for inelastic effects described by a number of internal variables D λm, and the CC network is considered to be purely elastic [1]. Accordingly, the free energy function of the rubber matrix can be represented as…”

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In this contribution, the previously proposed network evolution model [1] for carbon black filled elastomers is further studied. First, we show that the model does not contradict the second law of thermodynamics and is thus thermodynamically consistent.

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Micromechanical model for isolated polymer-colloid clusters under tension

“…The entropic force of a chain with N segments and the normalized end-to-end distancē r is written by F (r,N) = g(¯r N ) (see, e.g., Ref. [87]). Since the entropic forces at both parts of the backbone chain are identical, one has¯r 1 N 1 =¯r 2 N 2 .…”

Micro‐mechanical study of interactions between polymers and filler aggregates

“…Further, L −1 (x) denotes the inverse Langevin function while H represents the flexibility of the backbone chain [1]. Finally, T stands for the temperature (isothermal condition is assumed) while K is Boltzmann's constant.…”