Nonlinear region

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. 

Linear viscoelasticity Linear viscoelastic theory and its application to static stress analysis is now developed. According to this theory, material is linearly viscoelastic if, when it is stressed below some limiting stress (about half the short-time yield stress), small strains are at any time almost linearly proportional to the imposed stresses. Portions of the creep data typify such behavior and furnish the basis for fairly accurate predictions concerning the deformation of plastics when subjected to loads over long periods of time. It should be noted that linear behavior, as defined, does not always persist throughout the time span over which the data are acquired i.e., the theory is not valid in nonlinear regions and other prediction methods must be used in such cases. 

PREPARATIVE LAYER CHROMATOGRAPHY AS A PRACTICAL USAGE OF THE NONLINEAR REGION... 

Recently, these authors have treated specific adsorption at corrugated liquid-liquid interfaces [39] however, because of the complicated mathematics involved this work does not fall under the heading simple models. Also, Urbakh et al. [40] have extended the results for the interfacial capacity to the nonlinear region. 

Composite 1 with the higher fiber/matrix bonding shows a linear-elastic behavior up to = 160 MPa, which is clearly higher than that of composite 2. The following nonlinear region is small, resulting in an elongation at fracture of 0.32 %. 

A note of caution should be sounded here. Whilst the curves shown in Figure 6.5 are characteristic of many charged dispersions it should be recalled that once we apply a sinusoid to a non-linear system the response need not be a sinusoid. As the strain is increased into the nonlinear region, the waveform passing through the sample becomes progressively distorted. The instrumental analysis in this case involves... 

Fig. 8.4 Plots of relative change in electrical resistance against tensile deformation of a CNT/epoxy composite (a) shows the various characteristics of the piezoresistivity of nanocarbon networks linear resistance change in the elastic regime, nonlinear region after inelastic deformation and the permanent electrical resistance drop due to plastic deformation (image adapted from [30]) ...

Pulse testing does not work well on processes that are highly nonlinear because the pulse tends to drive the process away from the steadystate into a nonlinear region unless the pulse height is made very small. 

Fig. 6.6 Demonstration of the displacement phenomenon that occurs when FAC experiments are conducted in the nonlinear region of the binding isotherm. Three isobaric ligands (m/z 359) are infused at micromolar concentration through a sorbitol dehydro-...

Fig. 6.7 A FAC-MS experiment for ligand ranking. Eight ligands for sorbitol dehydrogenase were infused, spanning a range of 2 pM to 8 nM [10]. All ligands were applied at 1 pM concentration, ensuring operation in the nonlinear region of the binding isotherm. Adapted with permission from Elsevier.

BMD approach can provide information on the nonlinear region of the dose-response. 

This is a Gaussian which rapidly broadens and moves away from b. When it reaches the nonlinear region another approximation has to be used. 

At high x, stationary flow was not attained. The stationary flow was not detected, perhaps, because of limitations imposed by the experimental procedure, This can be exemplified by Ref. 17> in which the stationary flow was not attained in the nonlinear region and which was detected only in Ref.21 at high degrees of extension. 

In the nonlinear region, far from the solution, diagonal methods make more rapid progress. 

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