Strain cycling

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)) ... 

FIGURE 22.12 Uniaxial stress-strain cycles of ethylene-propylene-diene monomer (EPDM) samples with 60 phr silica at different prestrains = 10%, 20%, 30%, 40%, and 50% (symbols) and fittings (lines) with the stress-softening model Equations 22.19-22.24. The fitting parameters are indicated. The assumed cluster-size distribution is also shown, which differs from the one in Equation 22.24. (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]. 

As the wetting front advances at speed U, the solid undergoes a strain cycle at a variety of frequencies, /, the local frequency depending on the distance of the element of solid from the contact line at the moment under consideration. The solid the furthest from the contact line, yet still perturbed by the presence of the three-phase line, is at a distance of ca. to and thus feels a strain cycle at frequency [//to. At the other extreme, near the lower cutoff at x = 8, the frequency is ca. [7/8. The latter frequency will be dominant, since it is in the direct vicinity of the three-phase line that the solid is strained the most. As a consequence, and using Eq. (10), we can define the rate at which work is being done as ... 

Elastomer Cut growth air at 37°C Uncut blood at 37°C strain at 50% failure Uncut N2 at 37°C Uncut air at 37°C fixed strain cycle 1 hr. failure strain... 

However, in the case ofnorbornenes as substrates (Equation 8.1) a syn addition can be provided by the preferred exo addition at the strained cycle [25, 26]. 

A series of six stress-strain cycles with a crosshead rate of 600 mm/min was applied to specimens having a parallel length of 25 mm and a cross-section of 1 x 4 mm2 on a tensile testing machine. The samples were continuously stretched in six hysteresis cycles up to 60% of their elongation at break values, as shown in Fig. 47. This procedure is an established one and widely practiced for elastomeric composites reinforced with fillers such as carbon black and silica, which tend to build a strong filler-filler network [83]. 

The range of frequency specified is between 1 and 5 Hz and the standard only covers strain cycles passing through zero, although prestrains could be applied. It is suggested that at least five test pieces should be tested at each strain and that usually it is desirable to test at a number of maximum strains. The strain on ring test pieces is calculated on the internal diameter (see Chapter 8, Section 5.1). The test is continued until complete failure of the test piece occurs and then the number of cycles recorded. 

Fig. 2.16 Results obtained from an oscillatory shear experiment on an/ps = 0.103 PS-PEP diblock (Okamoto et al. 1994a). (a) SAXS patterns obtained at four representative strain phases as shown in (b) (c) a model showing (110) and (110) planes that give rise to the four diffraction peaks in (a). The pattern at each phase was obtained by integrating over [ A + 0 A + 0,-], where 0 = 0, id2, ittH and A = 0.194tt. Each pattern represents an average over 80 strain cycles.

Figure 12.16 Evolution of stress versus time for different fatigue tests depending on the stress ratio RCT and the corresponding stress-strain cycles crm is the average stress.

Using polyurethanes as a spring is an application where there is a tension and a compression cycle. The ability of the polyurethane under stress-strain cycling to reach a steady state is important. The material must reach... 

Alternating fields cause domain walls to oscillate. At low fields the excursions of 90°, 71° or 109° walls result in stress-strain cycles that lead to the conversion of some electrical energy into heat and therefore contribute to the dielectric loss. When peak fields are sufficient to reverse the spontaneous polarization the loss becomes very high, as shown by a marked expansion of the hysteresis loop (Fig. 6.9). 

Here, denotes the cluster size in spatial direction fi of the main axis system and dR (e ) is the norm of the relative stress of the rubber with respect to the initial stress at the beginning of each strain cycle, where de l dt=0 ... 

The application of this normalized, relative stress in Eq. (32) is essential for a constitutive formulation of cyclic cluster breakdown and re-aggregation during stress-strain cycles. It implies that the clusters are stretched in spatial directions with deu/dt>0, only, since AjII>0 holds due to the norm in Eq. (33). In the compression directions with ds /dt<0 re-aggregation of the filler particles takes place and the clusters are not deformed. An analytical model for the large strain non-linear behavior of the nominal stress oRjU(eu) of the rubber matrix will be considered in the next section. 

Stress-Strain Cycles of Filled Rubbers in the Quasi-Static Limit... 

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.

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... 

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). 

It is important to note that stress softening is also present during dynamic stress-strain cycles of filled rubbers at small and medium strain. In particular, this can be concluded from the dynamic mechanical data of the S-SBR samples filled with 60 phr N 220 as shown in Fig. 48. In the framework of the above model, the observed shift of the center point of the cycles to smaller stress values with increasing strain amplitude or maximum strain and the accompanied drop of the slope of the hysteresis cycles can be related to a de-... 

Fig. 48 Dynamic stress-strain cycles obtained from uniaxial harmonic excitations at 1 Hz of the S-SBR samples filled with 60 phr N 220 at fixed pre-strain and various strain amplitudes, as indicated...

From the simulations of stress-strain cycles at small and medium strain it can be concluded that the model of anisotropic cluster breakdown and re-... 

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