Stress softening

It is demonstrated in Figure 22.11 that the quasi-static stress-strain cycles at different prestrains of silica-filled rubbers can be well described in the scope of the above-mentioned dynamic flocculation model of stress softening and filler-induced hysteresis up to large strain. Thereby, the size distribution < ( ) has been chosen as an isotropic logarithmic normal distribution (< ( i) = 4> ) = A( 3)) ... 

The model describes the characteristic stress softening via the prestrain-dependent amplification factor X in Equation 22.22. It also considers the hysteresis behavior of reinforced mbbers, since the sum in Equation 22.23 has taken over the stretching directions with ds/dt > 0, only, implying that up and down cycles are described differently. An example showing a fit of various hysteresis cycles of silica-filled ethylene-propylene-diene monomer (EPDM) mbber in the medium-strain regime up to 50% is depicted in Figure 22.12. It must be noted that the topological constraint modulus Gg has... 

FIGURE 22.11 Uniaxial stress-strain data (up-cycles) of solution-based styrene-butadiene rubber (S-SBR) samples with 60 phr silica at different prestrains s =100%, 150%, 200%, 250%, and 300% (symbols) and fittings (lines) with the stress-softening model Equations 22.19-22.24. The fitting parameters are indicated. The insert shows a magnification of the small-strain data. (From Kliippel, M. and Heinrich, G., Kautschuk, Gummi, Kunststojfe, 58, 217, 2005. With permission.)... 

In particular it can be shown that the dynamic flocculation model of stress softening and hysteresis fulfils a plausibility criterion, important, e.g., for finite element (FE) apphcations. Accordingly, any deformation mode can be predicted based solely on uniaxial stress-strain measurements, which can be carried out relatively easily. From the simulations of stress-strain cycles at medium and large strain it can be concluded that the model of cluster breakdown and reaggregation for prestrained samples represents a fundamental micromechanical basis for the description of nonlinear viscoelasticity of filler-reinforced rubbers. Thereby, the mechanisms of energy storage and dissipation are traced back to the elastic response of tender but fragile filler clusters [24]. 

After a test specimen of an elastomer has been subjected to repeated deformation, the modulus at some lesser deformation is lower than the initial value. The modulus after such stress softening has been termed the degraded modulus. See Mullins Effect. 

There are various test methods, one being the De Mattia Flex Test method which is suitable for rubbers that have reasonably stable stress-strain properties, at least after a period of cycling, and do not show undue stress softening or set, or highly viscous behaviour. The results obtained for some thermoplastic rubber should be treated with caution if the elongation at break is below,... 

Work done by L. Mullins on the prestressing of filler-loaded vulcanisates showed that such prestressing gives a stress-strain curve approaching that of an unfilled rubber. This work has thrown much light on so called permanent set and the theory of filler reinforcement. See Stress Softening. 

An important feature of filled elastomers is the stress softening whereby an elastomer exhibits lower tensile properties at extensions less than those previously applied. As a result of this effect, a hysteresis loop on the stress-strain curve is observed. This effect is irreversible it is not connected with relaxation processes but the internal structure changes during stress softening. The reinforcement results from the polymer-filler interaction which include both physical and chemical bonds. Thus, deforma-tional properties and strength of filled rubbers are closely connected with the polymer-particle interactions and the ability of these bonds to become reformed under stress. 

The most striking feature of the stress softening phenomenon of thermoelastoplastics is its complete reversibility under certain conditions, consisting in a reformation process in the stress-free state which involves healing of cracks and reaching the initial integrity of the hard phase 119,120,12 ). The reformation kinetics in stress-softened samples indicates that this process is controlled by diffusion. 

Fig. 10. Internal energy changes as a function of deformation for oriented LDPE (I) and stress softened thermo-elastoplastic polyurethanes with 50% (2) and 42 % (3) hard phase content and polyether-polyester block copolymer with 48% hard phase content (4). The dotted curves 1 and 2 represent intramolecular energy changes for the corresponding polymers119 ...

The Payne effect of carbon black reinforced rubbers has also been investigated intensively by a number of different researchers [36-39]. In most cases, standard diene rubbers widely used in the tire industry, bke SBR, NR, and BR, have been appbed, but also carbon black filled bromobutyl rubbers [40-42] or functional rubbers containing tin end-modified polymers [43] were used. The Payne effect was described in the framework of various experimental procedures, including pre-conditioning-, recovery- and dynamic stress-softening studies [44]. The typically almost reversible, non-linear response found for carbon black composites has also been observed for silica filled rubbers [44-46]. 

Fig. 3 Example of stress softening with successively increasing maximum strain for an E-SBR-sample filled with 80 phr N 339...

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... 

Stress Softening and Filler-Induced Hysteresis at Large Strain 5.2.1... 

So far, we have considered the elasticity of filler networks in elastomers and its reinforcing action at small strain amplitudes, where no fracture of filler-filler bonds appears. With increasing strain, a successive breakdown of the filler network takes place and the elastic modulus decreases rapidly if a critical strain amplitude is exceeded (Fig. 42). For a theoretical description of this behavior, the ultimate properties and fracture mechanics of CCA-filler clusters in elastomers have to be evaluated. This will be a basic tool for a quantitative understanding of stress softening phenomena and the role of fillers in internal friction of reinforced rubbers. 

In view of an illustration of the viscoelastic characteristics of the developed model, simulations of uniaxial stress-strain cycles in the small strain regime have been performed for various pre-strains, as depicted in Fig. 47b. Thereby, the material parameters obtained from the adaptation in Fig. 47a (Table 4, sample type C60) have been used. The dashed lines represent the polymer contributions, which include the pre-strain dependent hydrodynamic amplification of the polymer matrix. It becomes clear that in the small and medium strain regime a pronounced filler-induced hysteresis is predicted, due to the cyclic breakdown and re-aggregation of filler clusters. It can considered to be the main mechanism of energy dissipation of filler reinforced rubbers that appears even in the quasi-static limit. In addition, stress softening is present, also at small strains. It leads to the characteristic decline of the polymer contributions with rising pre-strain (dashed lines in... 

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