Stretching elastic energy

In what follows, a lipid bilayer is discussed, with or without modified lipids in its matrix and only in its liquid crystal state, so that it can be considered as a two-dimensional liquid. The mechanical properties can thus be characterized by stretching and bending elastic moduli. Let Sq be the area of a flat tension-free membrane. If AS is the change of S, the density of the stretching elastic energy and its tension cr(AS) is then defined as... 

Other forces can arise as a result of elastic strain on the growing film, which can be due to a surface-induced ordering in the first few layers that reverts to the bulk liquid structure at larger distances. This elastic energy is stored in intermolecular distances and orientations that are stretched or compressed from the bulk values by the influence of the substrate at short distances [7]. Similar phenomena are well known to occur in the growth of epitaxial layers in metals and semiconductors. 

The values of stretching constant E that Crowley quoted in his original paper were substantially underestimated. As a result, his estimates of were in better agreement with experiment. However, did not depend on E and reached the same subnormal value. It must be noted that critical thinning I — almost universally reaches 30-40% for various models of the elastic energy [7],... 

Eq. (7.43)]. Explain why the stretching caused by isotropic swelling increases the contribution to the modulus from each combined strand. Why can the elastic modulus be approximated by an elastic energy density ... 

Elastic deformations within the solid can also produce static friction between surfaces that would not otherwise interlock. Figure lb illustrates a system where the spacing between peaks on the top surface is stretched to conform to the bottom surface. This could occur at the scale of either macroscopic asperities [21] (Section Vll) or individual surface atoms (Section 111). The elastic energy required to displace each peak into an opposing valley must be compensated by the gain in potential energy due to interactions between the surfaces. This... 

For an isotropic body with Poisson s ratio v = 0.5 stretched uniaxially, the elastic energy change is given by [137] ... 

By putting these equations to zero, we can calculate the equilibrium lateral displacements. In other words, we can calculate the deformations of the elastic layers in the x and y directions, from the current local variations in film height. Furthermore, by taking the functional derivative of the stretching free energy with respect to the height of the film, we also obtain the pressure contribution, which acts on the fluid layer. 

We use functional derivatives to calculate the forces from the elastic energy, and it is worth giving an example of this process. We, therefore, include a derivation of the forces for the bending of an elastic plate in one dimension. The extension of this to two dimensions, along with the derivation of stretching forces, is achieved in a similar manner. The bending free energy associated with a elastic plate in one dimension is obtained from Landau and Lifshitz [35] to be of the form... 

When a solid is uniformly elastically stressed, all bonds in the material elongate and the work done by the applied stress is converted to elastic energy that is stored in the stretched bonds. The magnitude of the elastic energy stored per unit volume is given by the area under the stress-strain curve (Fig. 11.1a), or... 

The reduction of degradation enhancement due to orientation is better seen when samples are stretched and then the time to fail, under UV radiation, is recorded. The results are shown in Fig. 6 where one should notice the break in scale for the reference (non-oxldlzed) sample. There is a drastic decrease in failure time (F.T.) for low draw ratios 1 < X < 1.7. This can be attributed to stored elastic energy which makes the chemical bonds more reactive toward UV, even at low stress levels. As X increases and the polymer structure becomes more and more oriented, F.T. Increases steeply before reaching a plateau once the orientation process is more or less completed. If we consider that photooxidation is oxygen diffusion controlled (1-5), the orientation effect is to decrease such diffusion by making the structure much more compact so that the degradation will be reduced. 

A close look at Fig. 14 shows that there is good correspondence between deformation regions and oxidation stages. First, there is an increase in oxidation which can be attributed to stored elastic energy. In the second stage, 1.7 < A < 3, since there is an "optically balanced state", i.e., the structure of the stretched film is very similar to that of the non-deformed one, we expect the samples to show similar oxidation content. 

The coefficient 6 accounts for the relative increase in elastic constants at the interface as compared to their bulk values. We shall then consider how independently perturbing the bending and stretching constants affects the elastic energy. 

It is clear that perturbation of the stretching constant (increase in 9g) has a much greater effect on the elastic energy barrier, than the increase in Ok- In fact, the perturbation of B alone is responsible for about 80% of the total increase in the elastic barrier. Moreover, if only the bending constant were affected, than the threefold increase in Ejnin would require Ok 40( ). These tendencies are illustrated in Fig. 8. 

Glass is a classic example atomic bonds can stretch (elasticity) or rupture, but typically do not reform. In metals, atomic bonds can slide and reform, a phenomenon described by dislocation theory this gives rise to energy dissipation, which makes it much more difficult to propagate a crack in metals. 

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