Dynamic strain frequency

Dynamic IR linear dichroism (DIRLD) studies have been made using a rapid-scan interferometer system. For dynamic strain frequencies achievable are between 0.1 Hz and 10 kHz step-scan interferometers are useful. 

The dynamic Weissenberg number W can be calculated from data obtained by the strain frequency sweep measurement. It s the ratio of the elastic to the viscous shares in the measured gel and leads to an objective description of the sensoric properties, representing the basis for the standardization of pectins. 

Fig. 43 Effect of dynamic strain amplitude on storage modulus (a). Stress-strain behavior of CR/ EPDM blend in the absence and presence of nanoclay (b). For this experiment, tension mode was selected for the variation of the dynamic strain from 0.01 to 40% at 10 Hz frequency...

The four variables in dynamic oscillatory tests are strain amplitude (or stress amplitude in the case of controlled stress dynamic rheometers), frequency, temperature and time (Gunasekaran and Ak, 2002). Dynamic oscillatory tests can thus take the form of a strain (or stress) amplitude sweep (frequency and temperature held constant), a frequency sweep (strain or stress amplitude and temperature held constant), a temperature sweep (strain or stress amplitude and frequency held constant), or a time sweep (strain or stress amplitude, temperature and frequency held constant). A strain or stress amplitude sweep is normally carried out first to determine the limit of linear viscoelastic behavior. In processing data from both static and dynamic tests it is always necessary to check that measurements were made in the linear region. This is done by calculating viscoelastic properties from the experimental data and determining whether or not they are independent of the magnitude of applied stresses and strains. 

When a sinusoidal strain is imposed on a linear viscoelastic material, e.g., unfilled rubbers, a sinusoidal stress response will result and the dynamic mechanical properties depend only upon temperature and frequency, independent of the type of deformation (constant strain, constant stress, or constant energy). However, the situation changes in the case of filled rubbers. In the following, we mainly discuss carbon black filled rubbers because carbon black is the most widespread filler in rubber products, as for example, automotive tires and vibration mounts. The presence of carbon black filler introduces, in addition, a dependence of the dynamic mechanical properties upon dynamic strain amplitude. This is the reason why carbon black filled rubbers are considered as nonlinear viscoelastic materials. The term non-linear viscoelasticity will be discussed later in more detail. 

Another important point is the question whether static offsets have an influence on strain amplitude sweeps. Shearing data show that this seems not to be the case as detailed studied in [26] where shear rates do not exceed 100 %.However, different tests with low dynamic amplitudes and for different carbon black filled rubbers show pronounced effects of tensile or compressive pre-strain [ 14,28,29]. Unfortunately, no analysis of the presence of harmonics has been performed. The tests indicate that the storage (low dynamic amplitude) modulus E of all filled vulcanizates decreases with increasing static deformation up to a certain value of stretch ratio A, say A, above which E increases rapidly with further increase of A. The amount of filler in the sample has a marked effect on the rate of initial decrease and on the steady increase in E at higher strain. The initial decrease in E with progressive increase in static strain can be attributed to the disruption of the filler network, whereas the steady increase in E at higher extensions (A 1.2. .. 2.0 depending on temperature, frequency, dynamic strain amplitude) has been explained from the limited extensibility of the elastomer chain [30]. 

In many studies it is presumed that linear viscoelastic behaviour always occurs, but this is not the case for many reactive systems. Conventional experimental rheology utilizes a dynamic strain sweep, which examines the dynamic rheological response to varied strain amplitudes, at a fixed frequency. If the system shows an effect of strain amplimde on dynamic properties (such as G or G") the system is said to be exhibiting a non-linear (viscoelastic) response. If the properties are independent of strain amplitude, then the system is said to be exhibiting linear viscoelastic behaviour. Figure 4.2 shows the response of an industrial epoxy-resin moulding compound (approximately 70 wt.% silica) at 90 °C at strain amplitudes of 0.1% to 10% for frequencies of 1, 10 and lOOrad/s. 

The relationship between steady-shear viscosity and dynamic-shear viscosity is also a common fundamental rheological relationship to be examined. The Cox-Merz empirical rule (Cox, 1958) showed for most materials that the steady-shear-viscosity-shear-rate relationship was numerically identical to the dynamic-viscosity-frequency profile, or r] y ) = r] m). Subsequently, modified Cox-Merz rules have been developed for more complex systems (Gleissle and Hochstein, 2003, Doraiswamy et al., 1991). For example Doriswamy et al. (1991) have shown that a modified Cox-Merz relationship holds for filled polymer systems for which r](y ) = t] (yco), where y is the strain amplitude in dynamic shear. 

Rheological measurements were performed in shear using a stress controlled rheometer (Carri-Med CSL 100) operating in cone-plate geometry. Each sample is submitted successively to a first frequency sweep in range 10 3-40 Hz under 3% strain, to a creep and recovery test, and finally to a second frequency sweep identical to the first one. The dynamical strain amplitude (3%) and the value of the creep stress (chosen so as to keep the maximum strain below 10%) were set in order to remain within the linear viscoelasticity domain. Creep and creep recovery were recorded during 20 h and 80 h, respectively, times which allowed the steady state to be reached in all cases. A fresh sample was used for each solvent/temperature combination. 

The bulk modulus K (= /H, the reciprocal of the bulk compliance) can be measured in compression with a very low height-to-thickness (h/i) ratio and unlubricatcd flat clamp surfaces. In pure compression with a high h/t ratio, and lubricated clamps, the compressive modulus (= /D, the reciprocal of the compressive compliance) will be measured. Any intermediate hjt ratios will measure part bulk and part compressive moduli. Hence it is vital for comparing samples to use the same dimensions in thermal scans and the same h/t ratio when accurately isotherming and controlling static and dynamic strains and frequencies. 

Scan frequency. Ability to track dynamic strains. Standard systems work up to 1 kHz, which is enough for most mechanical vibration phenomena special systems may attain 500 kHz, needed to track impacts or elastic... 

Figure 1-3 shows a schematic diagram of a dynamic IR linear dichroism (DIRLD) experiment [20-25] which provided the foundation for the 2D IR analysis of polymers. In DIRLD spectroscopy, a small-amplitude oscillatory strain (ca. 0.1% of the sample dimension) with an acoustic-range frequency is applied to a thin polymer film. The submolecular-level response of individual chemical constituents induced by the applied dynamic strain is then monitored by using a polarized IR probe as a function of deformation frequency and other variables such as temperature. The macroscopic stress response of the system may also be measured simultaneously. In short, a DIRLD experiment may be regarded as a combination of two well-established characterization techniques already used extensively for polymers dynamic mechanical analysis (DMA) [26, 27] and infrared dichroism (IRD) spectroscopy [10, 11]. 

Frequency of dynamic strain / dynamic dichroism Spatial average... 

Dynamic modulus at various strains, frequencies, and temperatures. 

Most of the rubbers are deformed dynamically and specified dynamic properties are required. Therefore the effect of strain amplitude on the dynamic modulus was observed very intensively. The modulus of filled mbbers decreases with increasing applied dynamic strain up to intermediate amplitudes. A detailed smdy of the low frequency dynamic properties of filled natural rubber was carried out by Fletcher and Gent [71] and was later extended by Payne [72, 73]. In cyclic strain tests the shear modulus can be simply expressed as a complex modulus, G = G + iG" where G is the in-phase modulus and G" the out-of-phase modulus. The phase angle 8 is given by, tan 5 = G"/G. ... 

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