Deformation amplitude

A typical behavior of amplitude dependence of the components of dynamic modulus is shown in Fig. 14. Obviously, even for very small amplitudes A it is difficult to speak firmly about a limiting (for A -> 0) value of G, the more so that the behavior of the G (A) dependence and, respectively, extrapolation method to A = 0 are unknown. Moreover, in a nonlinear region (i.e. when a dynamic modulus depends on deformation amplitude) the concept itself on a dynamic modulus becomes in general not very clear and definite. 

Visco-elastic fluids like pectin gels, behave like elastic solids and viscous liquids, and can only be clearly characterized by means of an oscillation test. In these tests the substance of interest is subjected to a harmonically oscillating shear deformation. This deformation y is given by a sine function, [ y = Yo sin ( t) ] by yo the deformation amplitude, and the angular velocity. The response of the system is an oscillating shear stress x with the same angular velocity . 

The deformation amplitude is varied [y (t) = yo sin (cot)] at a constant angular velocity. The resulting storage modulus (G ) is plotted versus the strain. 

The dynamic stiffness is the ratio of force amplitude to deformation amplitude ... 

After checking the influence of various experimental parameters as well as the reproducibility of the measurements, a standard procedure was developed. The polymer samples are cooled quickly with liquid nitrogen to a temperature of -140 °C. After an isothermal period of 1 min, a frequency sweep is carried out for 1,3,5,10, 30,40,50 and 60 Hz with a deformation amplitude of 20 im. Afterwards the temperature is increased by 5 °C and after another isothermal period of 1 min, the next frequency sweep is carried out. This... 

It demonstrates the highly elastic behaviour. At small deformation amplitudes the storage modulus G is one order of magnitude larger than the loss modulus G" and independent of the frequency. This is the behaviour of a solid body. 

The temperature dependence of the Payne effect has been studied by Payne and other authors [28, 32, 47]. With increasing temperature an Arrhe-nius-like drop of the moduli is found if the deformation amplitude is kept constant. Beside this effect, the impact of filler surface characteristics in the non-linear dynamic properties of filler reinforced rubbers has been discussed in a review of Wang [47], where basic theoretical interpretations and modeling is presented. The Payne effect has also been investigated in composites containing polymeric model fillers, like microgels of different particle size and surface chemistry, which could provide some more insight into the fundamental mechanisms of rubber reinforcement by colloidal fillers [48, 49]. 

The conclusion is that Lodge s rheological constitutive equation results in relationships between steady shear and oscillatory experiments. The limits y0 0 (i.e. small deformation amplitudes in oscillatory flow) and q >0 (i.e. small shear rates) do not come from Lodge s equation but they are in agreement with practice. These interrelations between sinusoidal shear deformations and steady shear flow are called the relationships of Coleman and Markovitz. 

A similar approach has also been developed by Susteric [108], who compares the behavior during low-amplitude deformation of rubbers, loaded with aggregated carbon black, with the visco-elastic behavior of macromolecules undergoing high-frequency deformation. The specific features of the breaking of carbon black aggregates defined by the deformation amplitude of loaded rubbers are described by the above author by a mathematical model developed for the description of the dynamic, visco-elastic behavior of polymer molecules. This approach revealed... 

The filler network break-down with increasing deformation amplitude and the decrease of moduli level with increasing temperature at constant deformation amplitude are sometimes referred to as a thixotropic change of the material. In order to represent the thixotropic effects in a continuum mechanical formulation of the material behavior the viscosities are assumed to depend on temperature and the deformation history [31]. The history-dependence is implied by an internal variable which is a measure for the deformation amplitude and has a relaxation property as realized in the constitutive theory of Lion [31]. More qualitatively, this relaxation property is sometimes termed viscous coupling1 [26] which means that the filler structure is viscously coupled to the elastomeric matrix, instead of being elastically coupled. This phenomenological picture has... 

A very convincing piece of evidence for agglomeration-deagglomeration as an important loss mechanism is the close empirical relationship between the maximum value of the loss modulus, G"m, and the height of the step in the storage modulus G (y0) as a function of deformation amplitude ... 

The same expression is obtained for the shear modulus G at small deformations under application of a shear stress oji. Conventional measurements of the cross-link density are, therefore, performed by defined sample deformation, for instance, in a rheometer during the vulcanization process of a test sample. The maximum rheometer moment for a given deformation amplitude is a direct measure of the cross-link density. Another, but invasive method for measuring cross-link densities in unfilled elastomer samples is by swelling in chloroform or toluene. In practice, spatial variations in cross-link density... 

Figure 9. Dependence of plastic deformation on deformation amplitude as a function of the number of deformation cycles.

FIGURE 8.18. Thermomechanical instability of a thin (110) Pt catalyst during oscillatory CO oxidation [52]. (a) Catalyst and support geometry, (b) Deformation amplitude as a function of time, (c) Images (4.4 x 4.4 mm ) from the catalyst s surface at various times marked in (b). 

Static tactile Stimulation that is a slow local mechanical deformation of the skin. It varies the deformation amplitude directly rather than the amphtude of vibration. This is normal touch for grasping... 

In 1949, Deuel and Neukom [131] sug sted that the origin of cro links in aqueous poly(vinyl alcohol)/borate gels is the complexation shown in Fig. 48. This concept of didiol type complexation was adopted by authors in more recent times [130,184-193]. Schultz and Myers [185] measured the dynamic moduli G and G" of polyfvinyl alcohol)/borate gels (at small deformation amplitudes in order not to disturb the gel [185]). At low frequendes, the gels showed Uquidlike behaviour this means that the borate crosslinks are dynamic in nature. Such results were also reported by Beltman [7]. He measu the dynamic moduli of a gel formed in a 4 wt% PVA98.5 solution with 2.5 wt% of borax (=0.0069 mol/1, which is equivalent to 0.0276 mol/1 of borate) at various temperatures (15-75 °C). Results are shown in Fig. 49. 

A further distinction between both types of network is the difference in ability to withstand deformation forces the gels obtained from HDPEl and 2 cannot withstand deformation amplitudes higher than 0.4% without being disturbed, whereas this amplitude is approximately 2% for the UHMWPE networks. 

Fig. 120. Dependence of the storage moduli of the gels mentioned in Fig. 119 vs composition of solvent angular frequency 0=1 rad/s deformation amplitude 0.5% (A) storage modultis (V) loss modulus. Reproduced, with permission of the author, from [327]...

There exists a related but different German Standard DIN 53 442 which uses dumb-bell-shaped specimens differing from those used for tensile testing by a rounded middle section. Another difference in comparison with the above ASTM method is the use of constant deformation amplitude of the vibrations. This results in a stress amplitude decreasing with time due to stress relaxation. Apart from this, the stress amplitude diminishes also due to the heating of the specimen. The results are reported in a similar manner as required by the ASTM standard with the stress amplitude relating to the first cycle. 

In the following, we expect an Arrhenius-like temperature behavior for highly filled rubbers that is typically fotmd for polymers in the glassy state. Therefore, we measure—far above the polymer bulk glass transition temperature—the modulus G for small deformation amplitudes (0.2% in our case). This is depicted schematically in Fig. 36.10. One obtains a straight line of slope E /R by plotting log G (T) (or in the tensile mode log E (T)) vs. 1/T well above the bulk... 

The spectra of the dynamic mechanical relaxations were obtained using rectangular sample strips (typical dimensions being 8 x 40.6 mm) with a DMA analyser in flexion mode. The temperature dependance of the loss modulus was determined with at 0.1 Hz, between -150 and 130 °C, at a heating rate of 5 C/min and using a deformation amplitude of 1 mm. 

The isolator is effective at = 0.9 it reduces the deformation amplitude to 39% of the response without isolators. At cojeo = 0.1 or 3, the isolator has essentially no influence on reducing the deformation. 

In the case of repeated mechanical influences with a constant deformation amplitude, destruction (or structuring) will lead to a reduction (increase) in the stresses and, consequently, to a deceleration (acceleration) of the breakdown of the body. On the other hand, in the case of a system of influence with constant stress amplitude, destruction (structuring) will already lead to an increase (decrease) in the value of the deformation amplitude and, consequently, to an acceleration (deceleration) of breakdown. 

For 1, the deformation amplitude C < reaches its final value C< exponentially with a smaller time constant l/2fl ... 

During the dynamical fatigue, the variation of material resistance with the stress or deformation amplitude is usually determined by drawing of Wohler curves (variation of O vs. N ), Figure 3.336, [914]. On these curves, three distinct regions can be delimited ... 

Figure 5.25 demonstrates the dependences of RMS on the reduced liquid crystal anchoring energy at the substrate Uw = Wod/Kss, It can be seen that with the rise of Uw the curve h(o s) becomes smoother and its maximum shifts toward larger values of ujs- Note that for low anchoring energy, Uw 0.5, the deformation amplitude (0 )max drastically increases, i.e., the assumption of the small director deviation (5.84), (5.85) becomes... 

The splay and bend elastic moduli, Ku and 33, also play an important role in the determination of a liquid crystal layer sensitivity and resolution. The smaller the elastic moduli, the larger the director deformation amplitude at a given voltage, Vq, i.e., sensitivity increases. This effect is particularly evident for low Ku values. The decrease in Ku leads to a considerable improvement in the layer spatial resolution. On the other hand, it would be unreasonable to decrease the X33 value dramatically, since, in this case, the resolution, t max, becomes worse. Consequently, we come to the conclusion that the optimum sensitivity and resolution of a liquid crystal layer with a homeotropic alignment can be achieved at K33 < 10 dyne and the highest possible elastic moduli ratio, KzzjKu [157, 163]. 

We now move to the perturbed system. The underlying perturbation idea is the following If the space dimension is two, the unperturbed solution Xp x—c[t- - y/p)) represents a steadily traveling wave whose wavefront forms a straight line in the y direction. Let the wavefront be given a slow wavy deformation as in Fig. 4.3 a or b. Locally the wavefront is supposed to be almost parallel to the y direction everywhere, while the deformation amplitude itself need not be small. The situation here reminds us of a one-dimensional array of diffusion-coupled limit cycle oscillators whose phase profile shows a slow spatial variation (Fig. 4.3 c). We have seen that each local oscillator is not much disturbed from its unperturbed closed orbit as far as the long-time behavior is concerned, so that in the crudest approximation one could assume... 

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