Basic Definitions Concerning Orientation

Problem. Let a polymer fiber contain rod-shaped structural entities in an amorphous matrix with some preferential orientation. Let us assume that the rods are crystalline. Our interest is to study the crystalline structure of the rods. Instead of sharp hkl reflections we observe that each reflection is smeared over a spherical cap in solid angle. Thus the observed intensity is suitably expressed in polar coordinates 

The goal is, both to gather information on the orientation distribution g f, y/) of the rods, and to reconstruct the scattering intensity lopt (s) of the perfectly oriented rod. 

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This chapter is concerned with the definition and evaluation of molecular anisotropy in liquid crystal polymers. Molecular orientation is a central theme of the liquid crystal state however, its definition requires considerable care. The basic mathematical relationships for the description of molecular anisotropy are reviewed together with an assessment of the various orienting units ranging from the molecular envelope to particular covalent bonds. The different methods of measuring orientation parameters for liquid crystal polymer samples are compared. Particular emphasis is placed upon X-ray and electron scattering procedures which are able to provide structural as well as orientational information. The experimental methods are highlighted through the use of several examples. 

Since it is not possible to obtain strengths in all possible lamina orientations or for all combinations of lamina, a means must be established by which these characteristics can be determined from basic layer data. Theories of failure are hypotheses concerning the limit of load-carrying ability under different load combinations. Using expressions derived from these theories, it is possible to construct failure envelopes or, if in three dimensions, failure surfaces that represent the limit of usefulness of the material as a load-bearing component, i.e., if a given loading condition is within the envelope, the material will not fail. The suitability of any proposed criterion is determined by a number of factors, the most important of which has to do with the nature of the failure mode. As a result, it is important that proposed failure criteria be accompanied by a definition of material behavior. 

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