Elastic distortion

Prefabricated emits produced under this section of the Code shall withstand a pressure test without failure, leakage, distress, or distortion other than elastic distortion at a pressure equal to the test pressure of the system in which they are installed, either before installation or during the system test. When such units are to be installed in existing systems, they shall be pressure tested before installation, unless the complete system is pressure tested after installation. 

Fig. la. Atomic structure ofa two-dimensional nano-structured material. For the sake of clarity, the atoms in the centers of the crystals are indicated in black. The ones in the boundary core regions are represented by open circles. Both types of atoms are assumed to be chemically identical b Atomic arrangement in a two-dimensional glass (hard sphere model), c Atomic structure of a two-dimensional nanostructured material consisting Of elastically distorted crystallites. The distortion results from the incorporation of large solute atoms. In the vicinity of the large solute atoms, the lattice planes are curved as indicated in the crystallite on the lower left side. This is not so if all atoms have the same size as indicated in Fig. la [13]... 

Lombardi, V., Piazzesi, G., Ferenczi, M. A., Thirlwell, H., Dobbie, I., and Irving, M. (1995). Elastic distortion of myosin heads and repriming of the working stroke in muscle. Nature 374, 553-555. 

The physical origin of the internal displacements is more easily seen in a two-dimensional analogue of the three-dimensional lattice illustrated in F ig. 8-8 and treated in Problem 8-3. If wc imagine the bonds as freely rotating springs (setting the constant C, equal to zero), we see that the elastic distortion tends to pull the shaded atoms upward with respect to the others. These are conventionally thought of as positive internal displacements. Note that if in contrast there were... 

Figure 26 HRTEM of a bent carbon nanotube, showing elastic distortions. Note the regular buckling in the compressed region.

Bateman and coworkers report a comparative analysis of six different hydraulically powered compaction simulators manufactured by a variety of vendors (Table 1), In their round-robin study, they found that the compaction simulators were comparable when operated within a moderate compression stress range of 50-2(X)MPa. However, at higher pressures, correction factors needed to be applied because of elastic distortion and differences in loading characteristics of the hydraulic systems (5). These results are not surprising since like rotary tablet presses, compaction simulators are not perfectly rigid. Therefore, compaction simulators should be properly calibrated (including corrections for mechanical flexure and electronic noise) to ensure the collection of quality experimental data. 

For the purposes of the present discussion, one of our ambitions is to consider a class of second-phase microstructures in which the two phases are coherent. In these instances, the constraint of coherency carries with it an elastic penalty which results in elastic distortions in both the precipitate particle and the matrix material surrounding it. As a preliminary to the concrete mathematical investigation of such structures, we must first learn something of the elastic consequences of a single inclusion. 

As noted above the spirit of locality is that the energy is reckoned point by point with an energy density of the form W(e) = CijiciCijeia. We note that the domain of validity of a locality assumption of this kind is based upon measuring the wavelength of the relevant elastic distortions against the lattice parameter. As noted... 

The strategy adopted in the work of Wolverton (1999, 2000) follows in the tradition of the models introduced in chap. 6. In particular, a cluster Hamiltonian which includes the energy cost of elastic distortions is fitted on the basis of a series of first-principles calculations. The set of structures used to effect this fit include special quasirandom structures, a clever idea introduced in a way that allows periodic structures to mimic their random counterparts, supercells including Cu layers and a variety of short period superlattices. The resulting effective... 

For the case of anisotropic particles with a rod-like shape (of length L and diameter D) in a nematic LC phase it was shown that the typical free-energy costs AF due to the elastic distortions at the LC-NPs interface is roughly ... 

Thus, elastic distortions give rise to a term that is roughly of the Flory-Huggins type, enhancing the phase-separation tendency of the system. 

Point defects. Vacancies, i.e., vacant regular sites and the atoms in interstitial cavities, i.e., interstitials are point defects of LRC. There are no reasons to consider these defects to be unstable. Their stability is provided by the local ordering of surrounding atoms. The nonelastic displacements of surrounding atoms as a result of point defect formation are possible in places of strong, close-to-critical, elastic distortions of the LRC network. 

The energy scale of an elastic distortion around a particle is of order KR, where iC is a typical elastic constant of the nematic liquid crystal [19] and R is the radius of the particle. For a thermotropic liquid crystal, K is approximately 10 N, and for a colloidal particle, i is approximately one micron thus the energy scale is a few thousands k TyWhere is the Boltzmann constant and T the temperature. As a result, the entropy of the particles is negligible compared to the elastic interactions. Under these conditions, the structures formed due to attractive interactions remain stable against thermal fluctuations. 

Fig. 24. The distortion produced by a screw dislocation (as shown in Fig. 4) may be simulated by the elastic distortion of a cylindrical ring.

Chadi and Martin (1976) used essentially the same LCAO parameters that are given in the Solid State Table to obtain the energies at the two special points in the Brillouin Zone. They then redetermined the wave numbers of the special points for the distorted crystal and recalculated the energy. The elastic distortion which they used is a shear strain, in which there is no change in bond length to first order in the strain thus the radial force constant of Eq. (8-1) does not enter the calculation. That strain can be written as... 

The stratified structure of a smectic liquid crystal imposes certain restrictions on the types of deformation that can take place in it. A compression of the layers requires considerable energy - very much more than for a curvature elastic distortion in a nematic - and therefore only those deformations are easily possible that tend to preserve the interlayer spacing. Consider the smectic A structure in which each layer is, in effect, a two-dimensional fluid with the director n normal to its surface. Assuming the layers to be incompressible, the integral... 

Making use of (5.8.1), and assuming the layers to be incompressible, the free energy of elastic distortion may be written in the form... 

When a tangential force, T is applied, elastic distortion causes the centers of the spheres to be displaced perpendicular to their axis (Mindlin and Deresiewicz, 1953 Dobry et al., 1982) by an amount given in the equation below ... 

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