Quasi-static limit

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

Beside the consideration of the up-cycles in the stretching direction, the model can also describe the down-cycles in the backwards direction. This is depicted in Fig. 47a,b for the case of the S-SBR sample filled with 60 phr N 220. Figure 47a shows an adaptation of the stress-strain curves in the stretching direction with the log-normal cluster size distribution Eq. (55). The depicted down-cycles are simulations obtained by Eq. (49) with the fit parameters from the up-cycles. The difference between up- and down-cycles quantifies the dissipated energy per cycle due to the cyclic breakdown and re-aggregation of filler clusters. The obtained microscopic material parameters for the viscoelastic response of the samples in the quasi-static limit are summarized in Table 4. 

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... 

The approaches are divided into those which do not take frequency into account (except for relaxation phenomena) and therefore are "static" theories and those which take frequency into account, generally by a scattering approach. The latter, "dynamic" theories, when reduced to the low frequency or quasi-static limit, usually compare favorably with the static theories. Some approaches take multiple scattering into account and cannot be solved in closed form. These require elaborate computer number crunching techniques (5). 

The Kerner equation, a three phase model, is applicable to more than one type of inclusion, Honig (14,15) has extended the Hashin composite spheres model to include more than one inclusion type. Starting with a dynamic theory and going to the quasi-static limit, Chaban ( 6) obtains for elastic inclusions in an elastic material... 

Similarly, the effects of bubbles in viscoelastic materials were studied by preparing rubber samples containing microvoids. Microvoids were used in order to avoid the effects of bubble resonance and to compare theories in their quasi-static limit. 

A combination of the sequences in Table 1 with MAS is also necessary to average out the CSA interaction. Such a combination is not straightforward, given the possible interference effects from the simultaneous presence of both RF and MAS. Such problems forced the above sequences to be applied in the quasi-static limit, where the MAS period Tj. is much longer than the cycle time of the sequence... 

An important consideration in a CRAMPS experiment is the interference between the two averaging processes, i.e., does the physical rotation of the sample by MAS impair the performance of the multiple-pulse sequence, the latter having originally been designed for static samples. Indeed, a low vr, i.e., less than 3 kHz, is used in a conventional CRAMPS experiment, such that, to a first approximation, the sample can be considered to be static during each cycle of the multiple-pulse sequence. In this so-called quasi-static limit, the multiple-pulse sequence can be considered to take care of the... 

It is interesting to note that Eq. (1.377) and Eq. (1.379) coincide exactly with the square of Eq. (1.199) and Eq. (1.201), respectively, which describe the field-enhancements of a dielectric sphere in the quasi-static limit. This is a manifestation of the so called optical reciprocity theorem and it can be shown that it holds for arbitrary geometry [46] (see also Sec. 5.3.3). 

Moreover, by decreasing the shell thickness we can note (see Fig. 3.17) a red-shift of the piasmon peak and this in agreement with extinction spectroscopies measurements on Au core-shells [57, 73], In the quasi-static limit, where the size of the nanoshell is much smaller than the wavelength of light, the piasmon resonance energies are determined by the aspect ratio [78, 93]. As the aspect ratio is increased, the piasmon resonance shifts to longer wavelengths [94,95]. 

Finally, the whole system (molecule + metal nanoparticle) can be treated atomistically via TD-DFT or other quantum chemical methods. The interaction between the metal nanoparticle and the molecule are treated on the same foot as the intra-molecule and intra-nanoparticle ones. This method is therefore able to include much more than just the electrodynamics coupling, as it can include mutual polarization, chemical bonding, charge transfers (also in excited states). On the down side, at present this approach is limited to very small metal particles (a few tens of atoms, a few nm in size). Moreover, electrodynamics coupling is limited to the quasi-static limit, as standard molecular Hamiltonian includes only non-retarded Coulombic potential. Nevertheless, this method represents a fully ab initio approach to molecular plasmonics. 

However, it is important to remark that for more realistic cases, this overlap between absorption and enhancement is lost, and for complex shaped particles for particles arrays and even for spherical particles outside the quasi-static limit, the maximum absorption frequency of the plasmon and the maximum available total field may appear at quite different frequencies. This is shown for example in Fig. 5.3 for a silver spherical particles. 

We can do a step further and, in the quasi-static limit, express Ynr in terms of the molecular transition dipole. To this aim, let us apply Eq. (5.26) to the volume W comprised between the two dashed surfaces in Fig. 5.4. We get ... 

Let us first consider the quasi-static limit V 0. For X = 0, equation (4.8) reduces... 

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