Nearly circular fibers

Higher-order modes of nearly circular fibers 289... 

Fundamental modes 16-3 Higher-order modes 16-4 Nearly circular fibers... 

When solutions of directly acting choUnomimetics are applied to the eye (i.e., conjunctival sac), they cause contraction of the smooth muscle in two important structures, the iris sphincter and the ciliary muscles (Fig. 12.3). Contraction of the iris sphincter decreases the diameter of the pupil (miosis). Contraction of the circular fibers of the ciliary muscle, which encircles the lens, reduces the tension on the suspensory ligaments that normally stretch and flatten the lens, allowing the highly elastic lens to spontaneously round up and focus for near vision (accommodation to near vision). 

Although human hair fibers vary in cross-sectional shape, from nearly circular to elliptical, normalizing most elastic and other properties to fiber thickness can significantly reduce experimental scatter. Thickness is usually characterized as fiber diameter or cross-sectional area. Corrections to diameter for ellipticity are generally not employed. Hair fiber dimensions are also necessary to calculate fundamental elastic properties, and dimensional changes are often employed to follow the course of chemical reactions with hair. 

Cross-sections of the fibers perpendicular to the fiber axis are nearly circular. 

We ignore the small polarization corrections to P and Py, given by Eq. (13-11), because P f Py for isotropic, noncircular waveguides. This is an accurate approximation, provided the material anisotropy is not so minute as to be comparable to the small contribution of order due to the waveguide structure. The higher-order modes of the noncircular waveguide have the same form as the fundamental modes, except when the fiber is nearly circular, for reasons given in Section 13-9. 

Symmetry properties of higher-order modes on circular fibers Alternative representation for modes on circular fibers Fibers with nearly circular cross-sections... 

Fibers with nearly circular cross-sections... 

We discussed higher-order modes of fibers with nearly circular cross-sections in Section 13-9, and showed that the transverse electric field must take the forms given at the bottom of Table 13-1, page 288. These forms are in terms of the solutions of the scalar wave equation of Eq. (32-37) and unknown constants a+ and a. When the cross-section is exactly circular, the two solutions of Eq. (32-37) have the same propagation constant. However, if the cross-section is only near to circular, the two solutions of the scalar wave equation have similar but distinct propagation constants P and p , as discussed in Section 13-8. 

The second special case is an orthotropic lamina loaded at angle a to the fiber direction. Such a situation is effectively an anisotropic lamina under load. Stress concentration factors for boron-epoxy were obtained by Greszczuk [6-11] in Figure 6-7. There, the circumferential stress around the edge of the circular hole is plotted versus angular position around the hole. The circumferential stress is normalized by a , the applied stress. The results for a = 0° are, of course, identical to those in Figure 6-6. As a approaches 90°, the peak stress concentration factor decreases and shifts location around the hole. However, as shown, the combined stress state at failure, upon application of a failure criterion, always occurs near 0 = 90°. Thus, the analysis of failure due to stress concentrations around holes in a lamina is quite involved. 

The cross-sectional shape of fibers is more complex. Many natural polymer fibers have unique cross-sectional shapes. For example, the cross-section of dry cotton is kidney-shaped, while that of degummed silk is nearly a triangle. These cross-sectional shapes are controlled by genetic codes, and human has limited influence. However, the cross-sectional shapes of synthetic polymer flbers, inorganic flbers, and nanofibers can be manipulated by controlling the fiber formation processes. The cross-sectional shapes of these fibers range from circular to oval, triangular, dog bone, trilobal or multilobal, hollow, etc. 

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