Strain amplification factor

Let us consider a volume V of filled compound the overall number N of CB aggregates in this sample is given by  

With respect to Equation [3], in a cross section of the volume V, the number of aggregates is  

When the sample is strained in one direction x by an amount AX, the junctions across a section perpendicular to the stretching direction contribute to the overall stress, in addition to the contribution of the rubber matrix. With respect to Equation [6], it follows fhat, over the cross section, the junctions contribute to a force given by  

From Medalia s floq simulation (see Chapter 4, section 4.1.4, Equation 4.6), 

Over the cross section, there also a contribution from the matrix rubber alone, so that the resulting stiffness can be estimafed as  

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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... 

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... 

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). 

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 ... 

Figure 6-10. Mooney-Rivlin plots of natural rubber filled with MT carbon black Top set actual data without using the strain amplification factor. Bottom curves after reduction using the strain amplification factor, equation (6-95). [After L. Mullins and N. R. Tobin, J. Appl. Polym. Sci., 9, 2993 (1965), by permission of John Wiley Sons, Inc.]...

Figure 7.10 a) Strain amplification factor as a function of strain for Samples 1, 3, 5 and 7. b) Strain amplification factor as a function of strain for Sample 4. 

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. 

The strain amplification factor is obviously depending on the filler loading and likely on the filler characteristics, firstly the structure. For a nonreinforcing CB (i.e., N990), which consists essentially of spherical particles with mean diameter of about 400 nm, Mullins and Tobin showed that an appropriate equation for the strain amplification factor is ... 

Up to this point, the strain amplification factor can be viewed as a mere empirical approach to assign the modulus increase in CB filled compound to filler level. Equation 5.19 above essentially resulted from considerations on the hydrodynamic effects induced by the presence of solid particles ideally dispersed in a matrix with a considerably lower modulus. The empirical factor f in Equation 5.20 adds nothing in this respect and it is well known that both equations do not suit at all either highly loaded compounds, whatever is the grade of CB, or moderately loaded materials with high structure blacks. Over the last decades, several authors have developed theoretical considerations to model the likely effect of a so-called filler network structure and the associated energy dissipation process when filled compounds are submitted to increasing strain. 

A relationship for the strain amplification factor can be derived from the NJ theory by considering the analysis made by Gent et al. - of the behavior of a rubber volume bonded between two rigid spheres. Indeed when a compressive (or a tensile) force F provokes a small displacement Ax of one sphere with respect to the other, there is a compression (or tensile) stiffness which consists of two parts (1) the rubber layer compressed (or stretched) between the two spheres, (2) the restraints at the bonded surfaces of the spheres. Gent and Park developed the following theoretical equation, i.e. ... 

A5.1.3 Strain Amplification Factor from the Network Junction Theory... 

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