Mullins effect filled rubber

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). 

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. 

Beside the characteristic stress softening up to large strains (Mullins effect) as shown in Fig. 36.13, the model also considers the hysteresis behavior of reinforced rubbers (Payne effect). Obviously, since the sum in Eq. (36.10) is taken over stretching directions with ds/dt > 0, only, up-and down cycles are described differently. An example considering a fit of the hysteresis cycles of silica filled EPDM rubber in the medium strain regime up to 50% is shown in Fig. 36.14. For these adaptations an alternative form of the cluster size distribution has been assumed, which allows for an analytical solution of the integrals in Eqs. (36.9) and (36.10) ... 

Clement, F. Bokobza, L. Monnerie, L., On the Mullins Effect in Silica-Filled Polydimethylsiloxane Networks. Rubber Chem. Technol. 2001, 74, 847-870. 

Examination of the stress as a function of time during retraction at constant strain rate following extension, and in particular for several successive cycles of elongation and retraction, reveals some further complications. The stresses during retraction are smaller than would be calculated on the basis of equation 8, and are smaller during extension in the second cycle than in the first. (This phenomenon, known as the Mullins effect, is particularly evident in filled rubbers, but appears also in the absenee of filler. )... 

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. 

Clement F, Bokobza L, Monnerie L (2001) On the Mullins effect in silica filled polydimethyl-siloxane networks. Rubber Chem Technol 74 846-70... 

Ogden RW, Roxburgh DG (1999) A pseudo-elastic model for the Mullins effect in filled rubber. Proc R Soc A 455 2861-2877... 

The Mullins effect, which can be considered as a hysteretic mechanism related to energy dissipated by the material during deformation, corresponds to a decrease in the number of elastically effective network chains. It results from chains that reach their limit of extensibility by strain amplification effects caused by the inclusion of undeformable filler particles [24,25]. Stress-softening in filled rubbers has been associated with the rupture properties and a quantitative relationship between total hysteresis (area between the first extension and the first release curves in the first extension cycle) and the enei-gy required for rupture has been derived [26,27]. 

There is considerable evidence that all the hysteresis effects observed in these materials and most of the viscoelastic behavior can be caused by the time dependent failure of the polymer on a molecular basis and are not due to internal viscosity [1,2]. At near equilibrium rates and small strains filled polymers exhibit the same type of hysteresis that many lowly filled, highly cross-linked rubbers demonstrate at large strains [1-8]. This phenomenon is called the "Mullins Effect" and has been attributed to micro-structural failure. Mullins postulated that a breakdown of particle-particle association and possibly also particle-polymer breakdown could account for the effect [3-5]. Later Bueche [7,8] proposed a molecular model for the Mullins Effect based on the assumption that the centers of the filler particles are displaced in an affine manner during deformation of the composite. Such deformations would cause a highly non-uniform strain and stress gradient in the polymer... 

In this relation, 2C2 provides a correction for departure of the polymeric network from ideality, which results from chain entanglements and from the restricted extensibility of the elastomer strands. For filled vulcanizates, this equation can still be applied if it can be assumed that the major function of the dispersed phase is to increase the effective strain of the rubber matrix. In other words, because of the rigidity of the filler, the strain locally applied to the matrix may be larger than the measured overall strain. Various strain amplification functions have been proposed. Mullins and Tobin33), among others, suggested the use of the volume concentration factor of the Guth equation to estimate the effective strain U in the rubber matrix ... 

Within identical validity limits, Mullins and Tobin have shown that the stress-strain behavior of black-loaded rubber vulcanizates corresponds to the stress-strain response of pure gum vulcanizates multiplied by a suitable strain amplification factor X, which expresses the fact that the average strain supported by the rubber phase, is increased by the presence of filler. In other terms, the effective strain of the elastomer matrix X is given by X =X.xX, where X is the overall measured deformation of the filled material. 

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