Ratio deformation amplitudes

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

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]. 

Figure 4.11. Diagrammatic sketches of atomic lattice rearrangements as a result of dynamic compression, which give rise to (a) elastic shock, (b) deformational shock, and (c) shock-induced phase change. In the case of an elastic shock in an isotropic medium, the lateral stress is a factor v/(l — v) less than the stress in the shock propagation direction. Here v is Poisson s ratio. In cases (b) and (c) stresses are assumed equal in all directions if the shock stress amplitude is much greater than the material strength.

At high set-point amplitude ratios rsp=As/Af> when the sample deformation and hence ks are relatively low (light tapping), the phase shift at C0=(0o can be written as... 

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]. 

When the stress is decomposed into two components the ratio of the in-phase stress to the strain amplitude (j/a, maximum strain) is called the storage modulus. This quantity is labeled G (co) in a shear deformation experiment. The ratio of the out-of-phase stress to the strain amplitude is the loss modulus G"(co). Alternatively, if the strain vector is resolved into its components, the ratio of the in-phase strain to the stress amplitude t is the storage compliance J (m), and the ratio of ihe out-of-phase strain to the stress amplitude is the loss compliance J"(wi). G (co) and J ((x>) are associated with the periodic storage and complete release of energy in the sinusoidal deformation process. Tlie loss parameters G" w) and y"(to) on the other hand reflect the nonrecoverable use of applied mechanical energy to cause flow in the specimen. At a specified frequency and temperature, the dynamic response of a polymer can be summarized by any one of the following pairs of parameters G (x>) and G" (x>), J (vd) and or Ta/yb (the absolute modulus G ) and... 

It has been shown [11] that the degree of deformation and the amplitude of oscillation of a liquid droplet depend on the ratio of the dynamic force to the surface-tension force, which is given by the Weber number, We = 2rp — u /S. Here S is the surface tension of the liquid, p... 

Several methods, as described in the preceding section, have been used to acquire information relative to the onset of the initial localized plastic deformation under alternating loading. Figure 5 is a plot of reflected light intensity vs. cycles for a transparent PS sample tested at 21 Hz at a stress amplitude of 17.2 MPa The number of cycles, N., to initiate the craze, as determined from the first jump in intensity over background, is about 5,000 cycles and the cycles to fracture, N, is about 11,000. In later sections, the ratio of to Np which in this example is about 0.45, is shown to be a function of both stress amplitude and frequency. 

Although creep-compliance (Kawabata, 1977 Dahme, 1985) and stress-relaxation techniques (Comby et al., 1986) have been used to study the viscoelestic properties of pectin solutions and gels, the most common technique is small-deformation dynamic measurement, in which the sample is subjected to a low-amplitude, sinusoidal shear deformation. The resultant stress response may be resolved into an in-phase and 90° out-of-phase components the ratio of these stress components to applied strain gives the storage and loss moduli (G and G"), which can be related by the following expression ... 

For many fragile materials and research/practical questions, intermittent CM is preferred as the lateral forces are practically avoided. Thus, sample damage or deformation is circumvented. However, despite the absence of shear forces, too high amplitudes or too low setpoint ratios may lead to damage of the sample or the tip as well. 

In contrast to observations in polystyrene, we do not observe permanent bands our specimens exhibit no residual birefringence upon release from stress. Neither do we observe crazing before failure. However, the specimens do whiten just before failure when viewed edge-on, and this whitening disappears within a few seconds after fracture occurs. We think the oscillations in intensity we observe are likely to be due to incipient shear deformation which disappears after specimen failure. Unpublished results of other workers are reported (see References 11 and 14 in the present Reference 12) to be consistent with the idea that such bands should be difficult to observe in PMMA and in polycarbonate because of their lower draw ratios compared to polystyrene. Studies of an unfilled epoxy polymer (14) in cyclic tensile deformation indicate that shear bands do not remain after removal of stress until a threshold amplitude of deformation is exceeded. 

The stress can be similarly resolved into two components with one along the direction of OB and one leading the strain vector by 7t/2 rad. The ratio of the in-phase stress to the strain amplitude (maximum strain) is called the storage modulus. In a shear deformation experiment this quantity is labeled G (o). The ratio of the out-of-phase stress to the strain is the loss modulus G (w). 

Jazayeri and Li [41] developed up to the third order nonlinear analysis of a liquid sheet to determine the breakup length of the sheet A typical result of their solution for the surface deformation as a function of distance is shown in Fig. 3.9. This case is for the initial disturbance amplitude of 0.1, the Weber number of 40 and the gas-to-liquid density ratio of 10, which approximates the situation of liquid water in ambient air. The wave number of 0.02 is almost equal to the dominant wave number for the sinuous disturbance of the linear theory. It is seen that the surface wave grows in time, and maintains its sinuous character for the majority of its growth... 

Considerable information about elastic and viscoelastic parameters may be derived by measuring the response of a polymer to a small-amplitude cyclic deformation. Molecules perturbed in this way store a portion of the imparted energy elastically, and dissipate a portion in the form of heat (Ferry, 1970 Meares, 1965 Miller, M. L., 1966, pp. 243-253 Nielsen, 1962, Chapter 7 Rosen, 1971 Schultz, 1974, pp. 67-71 Williams, D. J., 1971), the ratio of dissipation to storage depending on the temperature and frequency. In dynamic mechanical spectroscopy experiments, a cyclic stress is applied to a specimen, and two fundamental parameters are measured the storage modulus E a measure of the energy stored elastically, and the loss modulus a measure of the energy dissipated. The loss modulus E" may be calculated as follows ... 

The shape fixity ratio Rf can be determined for quantification of the effect of programnfing. Rf describes the ability to fix the mechanical deformation, which has been appUed during the programming process, i.e., Rf is equal to the amplitude ratio of the fixed deformation to the total deformation (see Fig. 8) [4],... 

---