Amplification, strain

It is important to note here that the presence of rigid filler clusters, with bonds in the virgin, unbroken state of the sample, gives rise to hydrodynamic reinforcement of the mbber matrix. This must be specified by the strain amplification factor X, which relates the external strain of the... 

In the case of a preconditioned sample and for strains smaller than the previous straining < e max). the strain amplification factor X in Equation 22.20 is independent of strain and determined by = X( , niax))- For the first deformation of virgin samples it depends on... 

In this connection, Fig. 2 provides a qualitative illustration for interpreting modulus change of an elastomer upon filler blending 9). A hydrodynamic or strain amplification effect, the existence of filler-elastomer bonds, and the structure of carbon black 10) all play a part in this modulus increase. 

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 ... 

It follows that this strain amplification effect will be more important if a high structure carbon black is used. In this case, indeed, the real volume concentration of the filler will be significantly increased by the amount of occluded rubber trapped in the aggregates. At strains high enough, the occluded rubber, though anchored or... 

Figure 7. Strain amplification A plot of the strain amplification ratio er as a function of the load frequency for different load magnitudes. Strain amplification ratio is defined as the ratio of the hoop strain in the cell process membrane to the bone surface strain at the osteonal lumen, e is the strain on the whole bone s is the load on the whole bone. Previously published in You et al. (2001).

Cowin, S.C. and Weinbaum, S. (1998) Strain amplification in the bone mechanosensory system. American Journal of Medical Science 316 184-188... 

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... 

For filler reinforced rubbers, both contributions of the free energy density Eq. (35) have to be considered and the strain amplification factor X, given by Eq. (39) differs from one. The nominal stress contributions of the cluster deformation are determined by oAtfJ=dWA/dzA, where the sum over all stretching directions, that differ for the up- and down cycle, have to be considered. For uniaxial deformations E =e, E2=Ej= +E) m- one obtains a positive contribution to the total nominal stress in stretching direction for the up-cycle if Eqs. (29)-(36) are used ... 

Figure 45a-c shows an adaptation of the developed model to uniaxial stress-strain data of a pre-conditioned S-SBR-sample filled with 40 phr N220. The fits are obtained for the third stretching cycles at various prestrains by referring to Eqs. (38), (44), and (47) with different but constant strain amplification factors X=Xmax for every pre-strain. For illustrating the fitting procedure, the adaptation is performed in three steps. Since the evaluation of the nominal stress contribution of the strained filler clusters by the integral in Eq. (47) requires the nominal stress aR>1 of the rubber matrix, this quantity is developed in the first step shown in Fig. 45a. It is obtained by demanding an intersection of the simulated curves according to Eqs. (38) and (44) with the measured ones at maximum strain of each strain cycle, where all fragile filler clusters are broken and hence the stress contribution of the strained filler clusters vanishes. The adapted polymer parameters are Gc=0.176 MPa and neITe= 100, independent of pre-strain. According to the considerations at the end of Sect. 5.2.2, the tube constraint modulus is kept fixed at the value Ge=0.2 MPa, which is determined by the plateau modulus Gn° 0.4 MPa [174, 175] of the uncross-linked S-SBR-melt (Ge=l/2GN°). The adapted amplification factors Xmax for the different pre-strains ( max=l> 1-5, 2, 2.5, 3) are listed in the insert of Fig. 45a.

Here, Xx and Xa are the strain amplification factors at infinite and zero strain, respectively, and y is an empirical exponent. An adaptation of this empirical function, Eq. (53), to the Xmax-values obtained for the pre-strained samples shown in Fig. 45a delivers the parameters X0=11.5, X -1.21, and y=0.8, which are used for the simulation of the first stretching cycle shown in Fig. 45c. 

Figure 46a shows that a reasonable adaptation of the stress contribution of the clusters can be obtained, if the distribution function Eq. (55) with mean cluster size =25 and distribution width b= 0.8 is used. Obviously, the form of the distribution function is roughly the same as the one in Fig. 45b. The simulation curve of the first uniaxial stretching cycle now fits much better to the experimental data than in Fig. 45c. Furthermore, a fair simulation is also obtained for the equi-biaxial measurement data, implying that the plausibility criterion , discussed at the end of Sect. 5.2.2, is also fulfilled for the present model of filler reinforced rubbers. Note that for the simulation of the equi-biaxial stress-strain curve, Eq. (47) is used together with Eqs. (45) and (38). The strain amplification factor X(E) is evaluated by referring to Eqs. (53) and (54) with i= 2= and 3=(l+ ) 2-l.

Fig. 47 a Uniaxial stress-strain data in stretching direction (symbols) of S-SBR samples filled with 60 phr N 220 at various pre-strains smax and simulations (solid lines) of the third up- and down-cycles with the cluster size distribution Eq. (55). Fit parameters are listed in the insert and Table 4, sample type C60. b Simulation of uniaxial stress-strain cycles for various pre-strains between 10 and 50% (solid lines) with material parameters from the adaptation in a. The dashed lines represent the polymer contributions according to Eqs. (38) and (44) with different strain amplification factors... 

Fig. 47b), which impacts the slope of the stress-strain cycles. This softening effect results from the drop of the strain amplification factor Xmax with increasing pre-strain, which has been determined by an extrapolation of the adapted values, shown in the insert of Fig. 47a, with the power law approximation Eq. (53). 

Little information has been published on the question of how filler network structure actually affects the energy dissipation process during dynamic strain cycles. The NJ-model focuses on modeling of carbon black network structure and examination of the energy dissipation process in junction points between filler aggregates. This model was further developed to describe the strain amplification phenomenon to provide a filler network interpretation for modulus increase with increasing filler content. 

Other strain-amplification relationships have been proposed on simple geometrical considerations. 

The effects of HAF black on the stress relaxation of natural rubber vulcanizates was studied by Gent (178). In unfilled networks the relaxation rate was independent of strain up to 200% extension and then increased with the development of strain induced crystallinity. In the filled rubber the relaxation rate was greatly increased, corresponding to rates attained in the gum at much higher extensions. The results can be explained qualitatively in terms of the strain amplification effect In SBR, which does not crystallize under strain and in cis-polybutadiene, vulcanizates of which crystallize only at very high strains, the large increase in relaxation rate due to carbon black is not found (150). 

It is certain that the relaxation behavior of filled rubbers at large strains involves numerous complications beyond the phenomena of linear viscoelasticity in unfilled amorphous polymers. Breakdown of filler structure, strain amplification, failure of the polymer-filler bond, scission of highly extended network chains and changes in network chain configuration probably all play important roles in certain ranges of time, strain rate, and temperature. A clear understanding of the interplay of these effects is not yet at hand. 

In applying Eq. (29) to data on black-filled rubbers Harwood and Payne (210) found that the basic 2/3-power relationship was preserved and that a data reduction by carbon black loading was possible upon replacement of Hb by Hb/X, with X assigned the role of a strain amplification factor ... 

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