Strain amplification factor modeling

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.

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.

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. 

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... 

To describe the soft phase contribution, one needs to develop a hyperelastic model taking into account (i) rubber elasticity behavior (ii) strain amplification due to the trapped hard phase inclusions (iii) strain hardening as the chains approach their maximum extensibility. Typically, one could approximate these effects using an inverse Langevin function or its Fade approximation (see ref. [39], Chapter 11), and using a strain multiplication factor. Here, we use a somewhat simplified expression that retains most of the required features ... 

Finally, standard spectral ratio (SSR) and horizontal-to-vertical spectral ratio (HVSR) methods for the determination of the deposit fundamental frequencies are becoming increasingly popular not only in the research field but also in professional practice (i.e., for microzonation studies). Site amplification factors can, in fact, be inferred, at least in the linear strain range, using the SSR technique described by Field and Jacob (1993). HVSR amplifications obtained experimentally can, instead, be used to validate numerical model results (e.g., SESAME 2004 Lanzo et al. 2011). 

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