Mullins effect mechanisms

It was shown that the stress-induced orientational order is larger in a filled network than in an unfilled one [78]. Two effects explain this observation first, adsorption of network chains on filler particles leads to an increase of the effective crosslink density, and secondly, the microscopic deformation ratio differs from the macroscopic one, since part of the volume is occupied by solid filler particles. An important question for understanding the elastic properties of filled elastomeric systems, is to know to what extent the adsorption layer is affected by an external stress. Tong-time elastic relaxation and/or non-linearity in the elastic behaviour (Mullins effect, Payne effect) may be related to this question [79]. Just above the melting temperature Tm, it has been shown that local chain mobility in the adsorption layer decreases under stress, which may allow some elastic energy to be dissipated, (i.e., to relax). This may provide a mechanism for the reinforcement of filled PDMS networks [78]. 

The Italian authors have also studied the effect of the annealing and of the solvent evaporation rate on the morphology and on the mechanical properties of different Kratons114-117. Kraton 1102 exhibits a Mullins effect which increases with annealing. Films with thickness between 0.2 and 0.4 mm, cast from dilute solution (1%) in cyclohexane and methylethyl ketone (MEK) at different rates have been... 

So far the micro-mechanical origin of the Mullins effect is not totally understood [26, 36, 61]. Beside the action of the entropy elastic polymer network that is quite well understood on a molecular-statistical basis [24, 62], the impact of filler particles on stress-strain properties is of high importance. On the one hand the addition of hard filler particles leads to a stiffening of the rubber matrix that can be described by a hydrodynamic strain amplification factor [22, 63-65]. On the other, the constraints introduced into the system by filler-polymer bonds result in a decreased network entropy. Accordingly, the free energy that equals the negative entropy times the temperature increases linear with the effective number of network junctions [64-67]. A further effect is obtained from the formation of filler clusters or a... 

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... 

The nano modified PDMS systems discussed have properties that enable a more confident prediction of ageing trends. The properties of the carborane modified system change in a linear fashion with radiation dose as opposed to the nonlinear trend observed for conventional particulate filled PDMS. The silica and carbon nano tubular systems display simplified mechanical properties. The reduction in Mullins effect or move to a more linear, less complex stress strain behavior, allows increased accuracy in property measurements. 

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. 

Non-linear viscoelastic mechanical behaviour of a crosslinked sealant was interpreted as due to a Mullins effect. The Mullins effect was observed for a series of sealants under tensile and compression tests. The Mullins effect was partially removed after a mechanical test, when a long relaxation time was allowed, that is the modulus increased over time. Non-linear stress relaxation was observed for pre-strained filler sealants. Time-strain superposition was used to derive a model for the filled sealants. Relaxation over long periods demonstrates that the Mullins effect is caused by non-equilibrium with experimental conditions being faster than return to the initial state. If experiments were conducted over times of the order of a day there may be no Mullins effect. If a filled elastomer were only required to perform its function once per day then each response might be linear viscoelastic. 

Hanson, D. E. Hawley, M. Houlton, R. Chitanvis, K. Rae, P. Orler, E. B. Wrobleski, D. A., Stress Softening Experiments in Silica-Filled Polydimethylsiloxane Provide Insight Into a Mechanism for the Mullins Effect. Polymer 2005,46,10989-10995. 

Hanson DE, Hawley M, Houlton R, Chitanvis K, Rae P, Oder EB et al (2005) Stress softening experiments in silica-filled polydimethylsiloxane provide insight into a mechanism for the Mullins effect Polymer 46 10989-95... 

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]. 

To accomplish these goals systematically, a brief discussion of constitutive linearity and non-linearity is introduced in Section 2. In Section 3 the "stress softening" or "Mullins Effect" [2-8] is analyzed from a simple mechanical failure model of the composite material s microstructure and is seen to contain this type of nonlinear behavior. Also the p order Lebesgue norms [13,14] of the strain history are presented as being excellent memory measures of the strain history to use in the constitutive equations. In Section 4 the development is extended to include the general three-dimensional constitutive equation for isotropic materials. 

There have been various models and mechanisms proposed for the Mullins Effect . Bueche proposed a model based on chains failing due to physically non-homogeneous local deformations [7,8]. His... 

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