By Maurice Holt

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When the load is decreased, damaged clusters can re-agglomerate, but the strength of the ﬁllerﬁller bonds is reduced, compared to the virgin bonds of undamaged clusters. 34 H. Lorenz, J. Meier, and M. Kl¨ uppel Also, damaged clusters are softer and more elastically deformable. Cyclic breakdown (stress release) and re-agglomeration of soft clusters causes hysteresis. The following paragraph describes the stress contribution of the ﬁller clusters. We apply this to isochoric uniaxial loading in 1-direction which fulﬁls the symmetry conditions λ ≡ λ1 and λ2 = λ3 = λ−1/2 .

Results will be discussed in the frame of the physically well understood material parameters obtained from the ﬁtting procedures. 1 Sample Preparation To identify parameter for the micromechanical model and to verify the model in further simulations, measurements were carried out with unﬁlled and ﬁlled rubber compounds. e. butadiene rubber (BR, Buna CB 10), solution styrene butadiene rubber (S-SBR, VSL 50250) and ethylene-propylene-diene rubber (EPDM, Keltan 512), respectively. e. ZnO, stearic acid and a semi-eﬃcient cross-linking system (sulfur + accelerator CBS), and anti-ageing IPPD.

During a closed hysteresis cycle, this is the closed integral of σdε. For a thermodynamically reversible elastic material law, as employed for the rubber matrix, the integral of the up cycle equals the negative of the down cycle, which means no production of entropy. As we will see later, the stress contribution of the ﬁller is always positive in the up-cycle and negative in the down cycle, which means that the spent mechanical work is always positive, and so is the entropy production. To describe the hyperelastic behavior of the rubber matrix, we use the nonaﬃne tube model with non-Gaussian extension [3–6] which has the following form: ⎫ ⎧ 3 Te 2 3 ⎬ ⎨ λ − 3 1 − μ μ=1 ne GC Te λ2μ − 3 + ln 1 − WR = 3 ⎭ 2 ⎩ 1 − Te ne λ2 − 3 ne μ=1 μ μ=1 3 λ−1 μ −3 .