Carbon-black-filled rubber Mullins effect

Govindjee, S. and Simo, J. (1991) A micro-mechanically based continuum damage model for carbon black-filled rubbers incorporating Mullins effect. Journal of the Mechanics and Physics of Solids, 39, 87-112. 

Roozbeh D, Mikhail I (2009) A network evolution model for the anisotropic Mullins effect in carbon black filled rubbers. Int J Solids Struct 46(16) 2967-2977... 

The behavior of carbon black-filled rubber in relation to quasi-static and dynamic responses is examined in detail. In particular, the main feamres of the microstructure of the material and their influence on the macro-mechanical response are highlighted. The effects of strain, strain-rate and temperature on the constitutive response are discussed. Mullins and Pa5me effects, which are peculiar in the behavior of filled elastomers, are reviewed and new results are shown. 

The above interpretations of the Mullins effect of stress softening ignore the important results of Haarwood et al. [73, 74], who showed that a plot of stress in second extension vs ratio between strain and pre-strain of natural rubber filled with a variety of carbon blacks yields a single master curve [60, 73]. This demonstrates that stress softening is related to hydrodynamic strain amplification due to the presence of the filler. Based on this observation a micro-mechanical model of stress softening has been developed by referring to hydrodynamic reinforcement of the rubber matrix by rigid filler... 

Figure 6-11. Stress softening of natural rubber filled with MPC carbon black (Mullins effect). Numerals indicate the stress-strain cycles. [After F. Bueche, J. Appl. Polym. Sci., 4, 107 (1960) by permission of John Wiley Sons.]...

Equation (1.5) has also been used to estimate the force at which a rubber molecule will become detached from a particle of a reinforcing filler (e.g., carbon black) when a filled rubber is deformed (Bueche, 1960, 1961). In this way, a general semiquantitative treatment has been achieved for stress-induced softening (Mullins effect) of filled mbbers (shown in Figure 1.5). 

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