Angle, orientation

In this paper, computations are performed for sizing of surface cracks with different orientations with respect to the applied field, complex cross-sections, and unknown shapes by using the algoritlim from It is shown that the algoritlim allows to perform sizing of cracks with complex cross-sections independent of the shape of the crack for orientation angles not exceeding 45°. 

Table 1. Computed results from the algorithm for sizing of the cracks from Fig.4a, Fig.4b, and Fig.4c and orientation angles d)=0°, 0=30° and 0=45°, and true values of the eorresponding crack parameters.

The data from Table 2 show that the algorithm developed in allows sizing of different cracks with complex cross-sections and unknown shapes for orientation angles not exceeding 45°. It is seen that the width 2a and the parameter c (or the surface density of charge m=4 r // e at the crack walls) are determined with 100% accuracy for all of the Case Symbols studied. The errors in the computation of the depths dj and di are less than 4% while the errors in the computation of d, dj, d, and d are less than 20% independent of the shape of the investigated crack and its orientation angle O <45°. 

For a laminate of N equal-thickness layers (N > 2) with orientation angles differing by ic/N as in Figures 7-53d and 7-53e, the summation for... 

At flow speeds well below the speed of sound, the lift coefficient depends only on the shape and orientation (angle of attack) of the body ... 

The stress curve sharply increases when the steric component appears upon compression. The initial thickness of a deformed layer is equal to be half the distance Dq obtained by extrapolating the sharpest initial increase to stress zero. The value Do is 21 1 nm, which is close the thickness of two molecular layers (19.2 nm) of the a-helix brush, calculated using the CPK model and the orientation angles obtained by FTIR analysis. We have calculated the elastic compressibility modulus Y,... 

In Eqs. (27) and (28), p is the contribution of the substrate water molecules, p that of the adsorbate polar head, and p that of the hydrophobic moiety of the adsorbed molecules. Consistently, 8i, 82, and 83 are the effective local permittivities of the free surface of water and of the regions in the vicinity of the polar head and of the hydrophobic group, respectively. The models have been used in a number of papers on adsorbed monolayers and on short-chain substances soluble in water. " Vogel and Mobius have presented a similar but more simplified approach in which p is split into two components only. " Recently some improvements to the analysis using Eq. (27) have been proposed. " An alternative approach suggesting the possibility of finding the values of the orientation angle of the adsorbate molecules instead of local permittivities has been also proposed.""... 

FIG. 2 Molecular orientation angles at liquid interfaces for rodlike molecules. The out-of-plane motion is a rotation away from the OZ axis, whereas the in-plane motion is performed with the OX, OY) plane. 

Molecules adsorbed at interfaces are in constant motion, both from the point of view of their orientation and of the dynamic exchange with the bulk solution. The distribution of orientation angle is characterized by the parameter D, see Eq. (13), or the angle of orientation Q. Complete randomness for the angle 0, the angle of rotation around the principal... 

Z = 4 D, = 1.52 R = 0.049 for 1,606 intensities. An error in the atomic coordinates for C-6 prevents display of this molecule. The atomic coordinates for C-l 1 were omitted, and those for H-61 and H-62 appear to be in error. The linkage, orientational angles between the hexenopyranosyl and the pyranoside ring are 0-5 -C-l -0-4-C-4 = +83°, C-l -0-4-C-4-C-5 = —139°. The epoxy ring is equilateral, with C-C = C-0 = 146 pm, and the ring-angles equal to 60 0.3°. The results of this analysis were correlated with those from the n.m.r. spectrum of the compound in solution. 

In order to simplify the discussion and to keep the derivation of the formulae tractable, the major part of this analysis is limited to a polymer fibre with a single orientation angle 0. This angle is assumed to be a kind of average angle and a characteristic parameter of the orientation distribution of the chain axes. 

The presented explanation for the existence of the fracture envelope will be used in formulating a fracture criterion for polymer fibres. Let us suppose a hypothetical polymer fibre with chains having a single orientation angle in the unloaded state. The shape of the fracture envelope is now calculated by taking into account the shear deformation of the chains only. For this case the work per unit volume up to fracture is given by... 

This simple fracture model has a major shortcoming. The exclusion of chain stretching in the model leads for small initial orientation angles to strength values that become infinite. It follows from Eq. 27 that the shear stress is a continuous function of the fibre stress and it increases asymptotically to the value of 2gtan . So for initial orientation angles... 

For a polymer fibre with a single orientation angle the modulus, E, or the slope at each point of the tensile curve, is a function of the tensile stress and given by... 

Equation 32 gives the total strain energy stored in a domain of a fibre with an orientation angle 0 in the unloaded state after the stress has been increased from 0 to o. The first term on the right-hand side is the strain energy of the chain extension, and the second term is the shear strain energy. The continu-... 

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